This blog post describes the rationale and motivation behind the LIBRARY OF WORDS, a digital collection of pages filled with every possible combination of 320 words.

ABSTRACT This document notes down expressions for a few common functions in terms of Hermite polynomials. There are clear symmetries between certain special functions, i.e. cos and sin, but interestingly a similar symmetry is formed between the Gaussian and the product of a Gaussian and the imaginary error function. The method of finding these expansions comes primarily from the Hermite transform. An approximate form for the log function is given, but this may have complex branching problems. MAIN Hermite Series Couplings: We can write a few common functions in a basis of Hermite polynomials e^{ax} = ^\infty }{2^n n!}H_n(x)\\ e^{x-{4}} = ^\infty {2^n n!}\\ e^{-x^2} = ^\infty {2^{2n}(2n)!}H_{2n}(x) \\ e^{-x^2} = \left[^\infty {2^{n}(n)!}H_{n}(x) \right] \\ ?(x) = \left[^\infty {2^{n}(n)!}H_{n}(x) \right] \\ \cos(ax) = \left[^\infty {e^{a^2/4}2^{n}n!}H_{n}(x)\right]\\ \sin(ax) = \left[^\infty {e^{a^2/4}2^{n}n!}H_{n}(x)\right]\\ x^n = {2^n} ^{\lfloor {2} \rfloor} {m!(n-2m)! } ~H_{n-2m}(x) where Hn(x) is a Hermite polynomial of order n. I want to find standard forms such that a function can be written f(x) = ^\infty c_n H_n(x) for some constants cn. It is interesting to see that there is a function marked as ?(x) above, which is made of the imaginary parts of the expansion of e−x². I would like to know what it is. We can see that ?(0)=0\\ ?(\infty)=0 and it has a single maximum just greater than x = 1. It seems we can write the Taylor expansion of this function as ?(x) = x - }{3}x^3 + }{15} x^5 - }{21} x^7 + \cdots \\ we can see then it looks like {} = ^\infty 2^{n-1}}{(2n-1)!!} x^{2n-1} \\ ?(x) = {2}}(x)e^{-x^2} For the logarithm, by experimental techniques we can formally write something like \log(x) = {2}(i\gamma + \pi + i\log(4)) + }{2^{2n-1}(2n-1)!}H_{2n-1}(x) + 24^{n-1}(n-1)!}{2^{2n}(2n)!}H_{2n}(x) but it’s not clear how it converges, perhaps something slightly wrong with this. Other functions use hypergeometric expressions as coefficients, for example \cosh() = ^\infty {4},{4};{256})}{2^n (2n)!}H_n(x) by setting x = 0, this gives us some sums for unity ^\infty {4},{4};{256})}{2^n (2n)!}}{\Gamma({2})} = 1 for the exponential expression ex − 1/4 given above we get ^\infty }{n!\Gamma({2})} = 1 also ^\infty {(2n)!\Gamma((1-n)/2)\Gamma(n+1/2)} \,_0F_4(;1/4 + n/2, 1/4+n/2,3/4+n/2,3/4+n/2;1/64) = 1 and {\pi^{5/2}}=^\infty }{(2n)!\Gamma({2}-n)} \,_0_4\left(;n+{4},n+{4},n+{4},n+{4};1/64\right) It seems that in general we have the result D_{2n-1} e^{-x^2} = ^\infty (2n+2k)!}{(2k+1)!(n+k)!}x^{2k+1} = {n!} \;_1F_1\left(n+{2},{2},-x^2\right) NICE RELATIONSHIP/TRANSFORM Using related functions above seems to make in interesting transform kernel, however, there does not seem to be a one to one mapping between transformed functions and input functions. Some results are given below: ^ \infty {8a}}J_0(x)}{} = I_0(a) ^ \infty {8a}}I_0(x)}{} = J_0(a) ^ \infty {8a}}e^x}{} = e^{3a} ^ \infty {8a}}e^x}{} = e^a ^ \infty {8a}}\cos(x)}{} = e^{-a} ^ \infty {8a}}\cos(x)}{} = e^{-3a} ^ \infty {8a}}\sin(x)}{} = 0 ^ \infty {8a}}\sin(x)}{} = 0 ^ \infty {8a}}J_2(x)}{} = I_1(a) ^ \infty {8a}}J_4(x)}{} = I_2(a)

ABSTRACT I introduce k-crossing paths and partitions and count the number of paths for each number of desired crossings k for systems with 11 points or less. I give some conjectures into the number of possible paths for certain numbers of crossings as a function of the number of points. INTRODUCTION A order n meandric partition is a set of the integers 1⋯n, such that a path from the south-west can weave through n points labeled 1⋯n without intersecting itself and finally heads east (examples are shown in Fig. 1). Counting the number of possible paths for n points is a tricky problem, and no recursion relation, generating function or explicit formula for the number of order n meandric partitions appears to have been found. This work is concerned with the number of paths that must intersect themselves exactly k times, where when k is 0, we have the meandric paths. It is possible to draw a line that deliberately crosses itself as many times as required, because of this we only consider a path to be k-crossing if k is the smallest number of crossings possible, that is a path that must cross itself k times (an example of a 3-crossing path over 9 points is given in Fig. 2). RESULTS Define ak(n) to be the number of configurations of n points where the path through them is forced to cross itself k times. For 0-crossings on n points we have the open meandric numbers, given in the OEIS as A005316 a_0(n) = 1, 1, 1, 2, 3, 8, 14, 42, 81, 262, 538, 1828, 3926, \cdots, \;\; n=0,1,\cdots this work has counted this for k > 0 by calculating all n! permutations of the n integers and checking to see the minimal number of crossings for each, we then have n =&0&1&2&3&4&5&6&7&8&9&10&11\cdots\\ a_0(n) =&1,& 1,& 1,& 2,& 3,& 8,& 14,& 42,& 81,& 262,& 538,& 1828,\cdots\\ a_1(n) =&0,&0,& 1,& 4,& 10,& 36,& 85,& 312,& 737,& 2760,& 6604, &25176,\cdots\\ a_2(n) =&0,&0,& 0,& 0,& 8,& 42,& 168,& 760,& 2418,& 10490,& 30842, &131676,\cdots\\ a_3(n) =&0,&0,& 0,& 0,& 2,& 16,& 164,& 944,& 4386,& 22240,& 83066, &398132,\cdots\\ a_4(n) =&0,&0,& 0,& 0,& 1,& 18,& 146,& 1076,& 6255,& 37250,& 168645, &908898,\cdots\\ a_5(n) =&0,&0,& 0,& 0,& 0,& 0,& 96,& 960,& 7388,& 51968,& 282122, &1711824, \cdots\\ a_6(n) =&0,&0,& 0,& 0,& 0,& 0,& 30,& 440,& 6472,& 55140,& 384065, &2642444,\cdots\\ a_7(n) =&0,&0,& 0,& 0,& 0,& 0,& 14,& 368,& 5176,& 53920,& 455944, &3575040,\cdots\\ a_8(n) =&0,&0,& 0,& 0,& 0,& 0,& 2,& 66,& 3542,& 45960,& 484058, &4336734,\cdots\\ a_9(n) =&0,&0,& 0,& 0,& 0,& 0,& 1,& 72,& 2011,& 32280,& 452504, &4661756,\cdots\\ a_{10}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 1172,& 25066,& 396493, &4709856,\cdots\\ a_{11}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 420,& 11840,& 309696, &4291440,\cdots\\ a_{12}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 201,& 8930,& 225754, &3661348,\cdots\\ a_{13}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 40,& 2240,& 151849, &2947392,\cdots\\ a_{14}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 18,& 2040,& 91147, &2103648,\cdots\\ a_{15}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 2,& 224,& 55030, &1575744,\cdots\\ a_{16}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 1,& 270,& 26762, &915924,\cdots\\ a_{17}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 14627, &665088,\cdots\\ a_{18}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 5405, &295956,\cdots\\ a_{19}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 2642, &218508,\cdots\\ a_{20}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 641, &63522,\cdots\\ a_{21}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 293, &54672,\cdots\\ a_{22}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 48, &8964,\cdots\\ a_{23}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 22, &9552,\cdots\\ a_{24}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 2, &706,\cdots\\ a_{25}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 1, &972,\cdots where the vertical sum over columns of terms gives n!. CONJECTURES The above information has lead to a few conjectures. a_{n^2}(2n) = 1 this can be converted to words as, there is exactly one path through 2n points that crosses n² times. The partitions associated with these paths are (2,1)\\ (3,1,4,2)\\ (4,1,5,2,6,3)\\ (5,1,6,2,7,3,8,4)\\ (6,1,7,2,8,3,9,4,10,5) and a clear interlaced pattern can be seen (an example is given in Fig. 3). a_{n^2-1}(2n) = 2, \; n>1 a_{n^2-2}(2n) = 4n+2, \; n>2 a_{n^2-3}(2n) = 8n+8, \; n>3 a_{n^2}(2n+1) = 2(n+1)3^{n-1}, \; n>1

ABSTRACT By introducing a new form of metric tensor the same derivation for the electromanetic tensor Fμν from potentials Aμ leads to the dual space (Hodge Dual) of the regular Fμν tensor. There are additional components in the i, j, k planes, however if after the derivation only the real part is considered a physically consistent electromagnetic theory is recovered with a relabelling of $$ fields to $$ fields and vice versa.

ABSTRACT We develop a triplet operator system which encompasses the structure of quark combinations. Ladder operators are created. The constants β are currently being found.

Gaia is a European Space Agency (ESA) astrometry space mission, and a successor to the ESA Hipparcos mission. The main goal of the Gaia mission is to collect high-precision astrometric data (i.e. positions, parallaxes, and proper motions) for the brightest one billion objects in the sky. This data, complemented with G band, multi-epoch photometric and low resolution (lowers) spectroscopic data collected from the same observing platform, will allow astronomers to reconstruct the formation history, structure, and evolution of the Galaxy. In addition, the Gaia satellite is an excellent transient discovery instrument, covering the whole sky (including the Galactic plane) for the next 5 years, at high spatial resolution (50 to 100 mas, similar to the Hubble space telescope (HST)) with precise photometry (1% at G=19) and milliarcsecond astrometry (down to ∼20mag). Thus, Gaia provides a unique opportunity for the discovery of large numbers of transient and anomalous events, e.g. supernovae, black hole binaries and tidal disruption events. We discuss the validation of the alerts stream for the first six months of the Gaia observations, in particular noting how a significant ground based campaign involving photometric and spectroscopic followup of early Gaia alerts is now in place. We discuss the validation approach, and highlight in more detail the specific case of Type Ia supernova (SNe Ia) to be discovered by Gaia. The intense initial ground based validation campaign will ensure that the Gaia alerts stream for the remainder of the Gaia mission, are well classified.

ABSTRACT Knowledge of genetic cause in neurodevelopmental disorders can highlight molecular and cellular processes critical for typical development. Furthermore, the relative homogeneity of neurodevelopmental disorders of known genetic origin allows the researcher to establish the subsequent neurobiological processes that mediate cognitive and behavioural outcomes. The current study investigated white matter structural connectivity in a group of individuals with intellectual disability due to mutations in _ZDHHC9_. In addition to shared cause of cognitive impairment, these individuals have a shared cognitive profile, involving oro-motor control difficulties and expressive language impairment. Analysis of structural network properties using graph theory measures showed global reductions in mean clustering coefficient and efficiency in the _ZDHHC9_ group, with maximal differences in frontal and parietal areas. Regional variation in clustering coefficient across cortical regions in _ZDHHC9_ mutation cases was significantly associated with known pattern of expression of _ZDHHC9_ in the normal adult human brain. The results demonstrate that a mutation in a single gene impacts upon white matter organisation across the whole-brain, but also shows regionally specific effects, according to variation in gene expression. Furthermore, these regionally specific patterns may link to specific developmental mechanisms, and correspond to specific cognitive deficits.

ABSTRACT I show how to express functions with a known series expansion in terms of Laguerre polynomials. MAIN We have the integral representation of the Laguerre polynomials, L_n(z)= {n!}\int_0^\infty e^{-t}t^n J_0(2)\;dt, where J₀(x) is a modified Bessel function, as given on the Wolfram Functions website. If we have a function f(x) that admits a series representation f(x)=^\infty a_k x^k, with some coefficients ak, then we can define a transform on L[f] F(s)=L[f(x)]=\int_0^\infty K(x,s)f(x)\;dx. If we pick the kernel function to be K(x,s)= e^{-x}J_0(2), then we have F(s) = ^\infty a_k\int_0^\infty e^{-x}J_0(2)x^k \; dx = ^\infty {e^s}L_k(s), which is an expansion of the transformed function F(s) in terms of Laguerre polynomials. As an example, if f(x)=1, then ak = [k = 0], this gives \int_0^\infty e^{-x}J_0(2)\;dx = ^\infty {e^s}L_k(s) = {e^s} = e^{-s} this leads to an infinitely recursive integral e^{-s} = \int_0^\infty \int_0^\infty \int_0^\infty \cdots \int_0^\infty e^{-x_n}J_0(2})\;dx_n \cdots J_0(2)\;dx_3J_0(2)\;dx_2J_0(2)\;dx_1 this should also lead to the set of results e^{-s} = \int_0^\infty e^{x_1}J_0(2)\;dx_1 e^{-s} = \int_0^\infty\int_0^\infty e^{x_2}J_0(2)J_0(2)\;dx_1dx_2 e^{-s} = \int_0^\infty\int_0^\infty\int_0^\infty e^{x_3}J_0(2)J_0(2)J_0(2)\;dx_1dx_2dx_3 and so on. If we formally define a function f(x) = ^\infty {e k!k!} = {e} I_0(2) we end up with a series expansion for F(s), starting F(s) = 1 - 2s + {4}s^2 - {18}s^3 + {576}s^4 - {7200}s^5 + \cdots It seems the inverse kernel is given by K^{-1}(x,s) = ^\infty ^i {(i-j)!j!j!} = ^\infty {i!}L_i(s) such that f(x) = \int_0^\infty K^{-1}(x,s)F(s)\;ds

ABSTRACT We investigate a function defined by a generalised continued fraction and find it to be closely related to a generating function of series OEIS:A000698. We then alter the function to contain the set of primes rather than the set of real numbers. This continued fraction function also has a similar form with coefficients 2, 6, 48, 594, 10212, 230796, ⋯. We also transform the sequence 1, 1, 1, 1, 1⋯ and gain the Catalan numbers as coefficients of the corresponding Laurent Series. We then provide examples which transform into various OEIS sequences. MAIN Define the function f(x)={x+{x+{x+\cdots}}} = {{\mathrm \large K \normalsize}} {x} evaluating this function until convergence for 16 decimal places, gives something that looks like 1/x, however, after diagnosing the coefficients of the Laurent series, by subtracting likely integer terms we very easily find f(x)={x}-{x^3}+{x^5}-{x^7}+{x^9}-{x^{11}}+{x^{13}}-\cdots, with a search on OEIS giving sequence A000698 which we believe to be plausible. We can easily imagine a similar function g(x)={x+{x+{x+{x+\cdots}}}} = {{\mathrm \large K \normalsize}} {x} where the top row of numbers are the prime numbers, pk. Using the first 9999 primes in the continued fraction, this also converges to 16 decimal places for large enough x, and appears to be described by an integral coefficient Laurent series g(x)={x}-{x^3}+{x^5}-{x^7}+{x^9}-{x^{11}}+{x^{13}}-\cdots That would allow conjecture for a very interesting relationship {{\mathrm \large K \normalsize}} {x} = {x} - {x^3} + {x^5} -{x^7} + {x^9} -\cdots where p₁ = 2,p₂ = 3 and so one for primes, and the capital K notation is that for the continued fraction. CONCLUSIONS A further assessment of many integer sequences inside the continued fraction should be made, and a check of corresponding Laurent series undertaken. It may be common place to find integral coefficients. The first function investigated in this document is also of interest, in OEIS the Laurent coefficient sequence A000698 is commented “Number of nonisomorphic unlabeled connected Feynman diagrams of order 2n-2 for the electron propagator of quantum electrodynamics (QED), including vanishing diagrams.” The continued fraction clearly has the potential to capture the recursive aspect of the theory, and this may be why the series align. ACKNOWLEDGMENT Many thanks to OEIS for providing their invaluable service. Thanks to Wolfram|Alpha for plots as below, and Mathematica for calculations. This work was undertaken in my spare time while being funded by the EPSRC.

MAIN We can write, for positive n, n^2=\exp\left(^1 {2-x}\;dx\right) we then have the product log interpretation of such an integral. TYPE I - GEOMETRIC INTEGRAL Let \prod_a^b f(x)^{dx}=f(x_i)^{\Delta x} then \prod_a^b f(x)^{dx}=\exp\left(\int_a^b \log f(x) \; dx\right) then we have n^2 = \exp\left({2-x_i}\right)^{\Delta x} let the smallest interval between the xi instead be 1, an integer discretization (ID of this. Then (n^2)= \exp\left({2-x_i}\right)^{\Delta x} = \exp\left(2\gamma +2\psi(0,n+1) \right) where for integer n \psi(0,n)=^{n-1}{k}-\gamma giving for integer n (n^2)= ^1\exp\left({2-i}\right)^{\Delta x} = \exp\left(^n {k} \right), \;n\in\\ (n^2)= ^1\exp\left({2-i}\right)^{\Delta x} = ^n\exp\left({k} \right), \;n\in then (n^2)}{n^2}=e^{2\gamma}\approx 3.17222 this then tells us that n^2 \approx {e^{2\gamma}}^n\exp\left({k} \right) but we also find a bias n^2 \sim {e^{2\gamma}}^n\exp\left({k} \right)-n-{3} and this relation holds very well for n > 10. This then provides the approximation ^n {k} \sim \log\left({3}}\right)+\gamma TYPE II - VOLTERRA PRODUCT INTEGRAL Let \prod_a^b (1+f(x)dx)=(1+f(x_i)\Delta x) then \prod_a^b (1+f(x)dx)=\exp\left(\int_a^b f(x)\;dx\right) then we have n^2=\left(1+{2-x_i}\right) and (n^2)=^1\left(1+{2-i}\right)={2} then {(n^2)}=2

ABSTRACT A p-rough number for some prime p, is a member of the set of numbers not divisible by any prime q < p. This document will note down the generating functions for the first few prime-rough numbers, and discuss their inverse Z-transforms, i.e a function a(n) that gives the nth member of the set. INTRODUCTION The 3-rough numbers are simply the odd numbers 1, 3, 5, 7, 9, 11⋯, the numbers which are not divisible by p ∈ {2}, See the table below for the appropriate OEIS label. {|c|c|} \hline p & p \\ \hline 2 & A000027 & 1,2,3,4,5,6,7\\ 3 & A005408 & 1,3,5,7,9,11,13\\ 5 & A007310 & 1,5,7,11,13,17,19\\ 7 & A007775 & 1, 7, 11, 13, 17, 19, 23\\ 11 & A008364& 1, 11, 13, 17, 19, 23, 29\\ 13 & A008365& 1, 13, 17, 19, 23, 29, 31\\ 17 & A008366& 1, 17, 19, 23, 29, 31, 37,\\ 19 & A166061& 1, 19, 23, 29, 31, 37, 41\\ 23 & A166063& 1, 23, 29, 31, 37, 41, 43\\ \hline From the table above we can see the first entry is always 1, the second entry is the nth prime, (i.e the index in the name of that sequence) and the next entry is the name of the next sequence. There are an increasing number of primes in each sequence, rather the probability of finding a prime number is higher. GENERATING FUNCTIONS A generating function for a sequence with terms a(n) is simply the series, G_a(z)=a(0)z^0 + a(1)z^1 + a(2)z^2 + \cdots= ^\infty a(n)z^n, however, for some sequences it makes more sense to count from 1, giving the description G_a(z)=a(1)z^1 + a(2)z^2 + \cdots= ^\infty a(n)z^n, the latter is the description we will use. We can see that the generating function for the natural numbers 1, 2, 3, ⋯ (which are also the 2-rough numbers) is then G_2(z)={(z-1)^2}=z+2z^2+3z^3+\cdots\\ a_2(n)=n Then for the 3-rough numbers we have the odd numbers, G_3(z)={(z-1)^2} \\ a_3(n) = 2n-1 we can use the Mathematica expression \\ to extract the sequence function a₃(n). For the 5-rough numbers we have numbers congruent to 1 or 5 mod 6, G_5(z) ={(1-z)^2(1+z)}\\ a_5(n) = {2}(6n+(-1)^n-3) From these we notice that G_3(z)=G_2(z)(1+z)\\ G_5(z)={G_3(z)}(1+4z+z^2) From here on things get less simple. The next sequence is the 7-rough numbers G_7(z)={(1+z)(z^2+1)(z^4+1)(z-1)^2} But the sequence description a(n) is terribly complicated a_7(n)= {8}e^{-{4}in\pi}\left((1-3i)+(2+i)+(-1)^n\left((1-3i)-(2+i)+((i-14)+30n)\cos(n\pi/4)+(2+6i)\cos(n\pi/2)+(2-i)\cos(3n\pi/4)+((16i-1)-30in)\sin(n\pi/4)+(2+4i)\sin(n\pi/2)-\sin(3n\pi/4)\right)\right) Then we have the 11-rough numbers G_{11}(z) = + 10z^{47} + 2z^{46} + 4z^{45} + 2z^{44} + 4z^{43} + 6z^{42} + 2z^{41} + 6z^{40} + 4z^{39} + 2z^{38} + 4z^{37} + 6z^{36} + 6z^{35} + 2z^{34} + 6z^{33} + 4z^{32} + 2z^{31} + 6z^{30} + 4z^{29} + 6z^{28} + 8z^{27} + 4z^{26} + 2z^{25} + 4z^{24} + 2z^{23} + 4z^{22} + 8z^{21} + 6z^{20} + 4z^{19} + 6z^{18} + 2z^{17} + 4z^{16} + 6z^{15} + 2z^{14} + 6z^{13} + 6z^{12} + 4z^{11} + 2z^{10} + 4z^9 + 6z^8 + 2z^7 + 6z^6 + 4z^5 + 2z^4 + 4z^3 + 2z^2 + 10z + 1)}{(z^{49} - z^{48} - z + 1)} \\ G_{11}(z) = +1 + 2(z^{46}+ z^{44}+ z^{41}+ z^{38}+ z^{34}+ z^{31}+ z^{25}+ z^{23}+ z^{17}+ z^{14}+ z^{10}+ z^7+ z^4+ z^2) + 4(z^{45} + z^{43} + z^{39} + z^{37} + z^{32}+ z^{29}+ z^{26} + z^{24} + z^{22} + z^{19} + z^{16} + z^{11}+z^9+z^5+z^3) + 6(z^{42} + z^{40} + z^{36} + z^{35} + z^{33} + z^{30} + z^{28} + z^{20} + z^{18} + z^{15} + z^{13} + z^{12} + z^8 + z^6) + 8(z^{27} + z^{21})+ 10(z^{47}+z) )}{(z^{49} - z^{48} - z + 1)} Whose sequence description takes us a few screens and is silly to place here... For this we have learnt that the generating functions remain somewhat simple compared to the sequence descriptions.

ABSTRACT This work strongly relates to Lambert series, but uses an alternative kernel in the transformation. Many results are shown from experimental techniques and tabulated below. MAIN From the product representation of ez we can write x = ^\infty {k} \log(1-x^k) or perhaps more nicely x = ^\infty {k} \log\left({1-x^k}\right) I then wonder what other functions can be written in terms of this, for example e^x -1 = ^\infty {k!}\log(1-x^k) where the ak starting from a₁ are −1, 0, 1, 5, 23, 59, 719, 839 which appear to be A253901 minus 1, an then we have e^x \approx 1+^\infty {k!}\left((d-1)!^{\mu\left({d}\right)}-1\right)\log(1-x^k) but this seems to be wrong for the 12th power and some others, so this is not the full story! We seem to have x^2 = ^\infty {k} \log(1-x^{2k}) in fact it seems x^n = ^\infty {k} \log(1-x^{nk}) which is fairly obvious from taking the power of the argument, however we could also have squared the entire sum, so we know that ^\infty {k} \log(1-x^{2k}) = \left(^\infty {k} \log(1-x^{k})\right)^2 and more generally ^\infty {k} \log(1-x^{nk}) = \left(^\infty {k} \log(1-x^{k})\right)^n interestingly we can write \log(1-x^m)= ^\infty ^\infty {nk}\log(1-x^{mnk}) and we must have the expansion e^x =1 -^\infty ^\infty {n!k}\log(1-x^{nk}) we can get from this the generating function of the Bell Numbers e^{e^x-1}= ^\infty ^\infty (1-x^{nk})^{{kn!}} of course this is all for x < 1. The product seems to converge for constants for example ee1/2 − 1. Other interesting results {(1-x)} = ^\infty ^\infty {k}\log(1-x^{nk}) {(1-x)^2} = ^\infty ^\infty -n{k}\log(1-x^{nk}) O\left[{\zeta(s-1)}\right] = ^\infty ^\infty {k}\log\left({1-x^{nk}}\right) ^\infty A101035(n){n} = ^\infty ^\infty \mu(n)\mu(k)\log\left({1-x^{nk}}\right) -\log(1-x) = ^\infty ^\infty \mu(n)\log\left({1-x^{nk}}\right)\\ -\log(1-x) = ^\infty ^\infty \mu(k)\log\left({1-x^{nk}}\right)\\ -\log(1-x) = ^\infty ^\infty {nk}\log\left({1-x^{nk}}\right)\\ -\log(1-x) = ^\infty ^\infty {nk}\log\left({1-x^{nk}}\right)\\ ^\infty d\tau(d)x^n = ^\infty ^\infty \log\left({1-x^{nk}}\right)\\ where O hypothetically maps a Dirichlet generating function to an ordinary generating function. The last one is a statement that ^\infty e^{ d \tau(d) x^n}= ^\infty ^\infty {1-x^{n k}} this has a few similarities to the Euler product of the zeta function \zeta(s) = } {1-p^{-s}} if we let x = 1/p, then we get ^\infty e^{ d \tau(d) p^{-n}}= ^\infty ^\infty {1-p^{-n k}} ^\infty e^{d \tau(d) p^{-n}}= ^\infty ^\infty {1-p^{-n k}} this then is a relation ^\infty e^{d \tau(d) p^{-n}}= ^\infty {({p^n};{p^n})_\infty} interestingly we could write ^\infty e^{d \tau(d) p^{-n}}= \left(^\infty {1-p^{-k}}\right)^\infty ^\infty {1-p^{-n k}} then take the product over all primes }^\infty e^{d \tau(d) p^{-n}}}{^\infty ^\infty {1-p^{-n k}}}= }\left(^\infty {1-p^{-k}}\right) }^\infty e^{d \tau(d) p^{-n}}}{^\infty ^\infty {1-p^{-n k}}}= ^\infty\zeta(k) of course this is probably nonsense OTHER INTERESTING SUMS we can change μ(n) to be something else for example we seem to have ^\infty^\infty {k}\log\left({1-x^{kn}}\right) = ^\infty d(n)x^n ^\infty^\infty {nk}\log\left({1-x^{kn}}\right) = ^\infty {n}x^n ^\infty^\infty {1}\log\left({1-x^{kn}}\right) = ^\infty {n}x^n for Euler totient function φ(n) and divisors d(n). ^\infty^\infty k\log\left({1-x^{kn}}\right) = ^\infty {n}x^n ^\infty^\infty \sigma_0(k)\log\left({1-x^{kn}}\right) = ^\infty {d})\tau(d)}{n}x^n ^\infty^\infty k^2\log\left({1-x^{kn}}\right) = ^\infty {d})}{n}x^n ^\infty^\infty {k}\log\left({1-x^{kn}}\right) = ^\infty {n}x^n other results are,k/n transforms to A007433,(k − 1)/k transforms to A069914/n. We seem to have ^\infty^\infty {n}\left(\mu(d)\mu\left({d}\right)\right)\log\left({1-x^{kn}}\right) = ^\infty {n}x^n ^\infty^\infty \left(\mu(d)\mu\left({d}\right)\right)\log\left({1-x^{kn}}\right) = ^\infty {n}\left(d \mu(d)\right)x^n ^\infty^\infty \lambda(n)\log\left({1-x^{kn}}\right) = ^\infty (n)}{n}x^n OTHER ^\infty^\infty \lambda_3(n)\log\left({1-x^{kn}}\right) = -\log(1-x)+^\infty x^{n^3} + ^\infty ^\infty }{n} where λ₃(n) is the equivalent of the Liouville function for squares but for cubes.

MAIN We can package a sum over n into a set of sums with a triangular number like stride. We can see this by taking the integers, removing the triangular numbers, then shifting by one, removing the triangular numbers again 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,\cdots\\ .,2,.,4,5,.,7,8,9,.,11,12,13,14,.,16,17,18,19,20,.,22,23,24,25,26,27,.,29,30,\cdots\\ .,.,.,.,5,.,.,8,9,.,.,12,13,14,.,.,17,18,19,20,.,.,23,24,25,26,27,.,.,30,\cdots\\ .,.,.,.,.,.,.,.,9,.,.,.,13,14,.,.,.,18,19,20,.,.,.,24,25,26,27,.,.,.,\cdots\\ and so on ^\infty f(n) = }f(n) + }f(n) + }f(n) + \cdots the first coefficients of the latter sums take the formula c_{1,0}=1\\ c_{1,n}={2}(3n+n^2) where we are calling the first sum 0 and the later sums 1, 2, 3, 4, ⋯, n, ⋯. Then we have that c_{n,0}={2}\\ c_{n,1}=1+{2} = 1+c_{n,0} \\ c_{n,2}=2+{2} = 1+c_{n+1,1} = 2+c_{n+1,0} \\ c_{n,3}=3+c_{n+2,0} \\ c_{n,4}=4+c_{n+3,0} \\ c_{n,m}=m+c_{n+m-1,0} = m + {2}, \;m>0 so we can rewrite ^\infty f(n) = ^\infty f(c_{n,0}) + ^\infty f(c_{n,1}) + ^\infty f(c_{n,1}) + \cdots\\ ^\infty f(n) = ^\infty ^\infty f(c_{n,m}) \\ ^\infty f(n) = ^\infty f\left({2} \right)+ ^\infty ^\infty f\left(m + {2}\right) Is this useful? It seems to complicate things alot, for example the normal example \zeta(2)=^\infty {n^2}={6} The first term is easy S_0=^\infty \left({2}\right)^{-2} = {3}(\pi^2-9) the rest is then S_1=^\infty {(1+8n)^{3/2}} \left(2(\psi_0(t_-)-\psi_0(t_+))+(\psi_1(t_-)+\psi_1(t_+)) \right) with ψn the polygamma function, and t_+= {2} + n +{2} \\ t_-= {2} + n -{2} we then must have that S₀ + S₁ = π²/6. So S_1={6}(72-7\pi^2) But we can also write a chain of expressions for the mth sum, {6} = {3}(4\pi^2-36) + {27}(4\pi^2-31) + S_2 + \cdots but some of these sums are hard to write here. We find the value indeed converges towards the required value of S₁ ≈ 0.485462. The sum is not particularly efficient. ZETA(3) We can work towards a sum for ζ(3) though! We find from similar treatment we have \zeta(3)= 8(10-\pi^2) - ^\infty {(1+8n)^{5/2}}\Bigg( 12(\psi_0(t_-)-\psi_0(t_+)) + 6( \psi_1(t_-) + \psi_1(t_+)) + (1+8n)(\psi_2(t_-)-\psi_2(t_+)) \Bigg) and the remaining terms converge on the correct number 0.15889211187406554. For triangular numbers, the summand simplifies to a nicer form. SUM ^\infty {2^n} = 1 so then when we perform ^\infty 2^{-(n(n+1)/2)} = {2^{7/8}}\theta_2(0,{})-1 ^\infty ^\infty 2^{-\left(m+{2}\right)} for m = 1 we get {2\cdot2^{7/8}}\theta_2(0,{})-{2} GENERATING FUNCTIONS We may write the set of powers of x as generating functions as follows T={2}\theta_2(0,) for elliptic theta, then -1+x^{-1/8}T = x + x^3 + x^6 + x^{10} + x^{15} + x^{21} + x^{28} + \cdots \\ -x + x^{7/8}T = x^2+x^4+x^7+x^{11}+x^{16}+x^{22} + \cdots\\ -x^2(1+x) +x^{15/8}T = x^5 + x^8 + x^{12} + x^{17} + x^{23} + x^{30} + \cdots \\ and the general term becomes -1+x^{-1/8}T, \;n=0 \\ -x^n\left(^{n-1} x^{k(k+1)/2}\right) + x^{(8n-1)/8}T = ^\infty x^{n+(k+n-1)(k+n)/2}, \;n>0 Thus {1-x} = -1+x^{-1/8}T + ^\infty -x^n\left(^{n-1} x^{k(k+1)/2}\right) + x^{(8n-1)/8}T \\ {1-x} = A + ^\infty -x^n\left(^{n-1} x^{k(k+1)/2}\right) where A= -1+x^{-1/8}T + ^\infty x^{(8n-1)/8}T\\ A = )}{(2-2x)x^{1/8}} -1 where A + 1 is the beautiful series 1 + 2x + 2x^2 + 3x^3 + 3x^4 + 3x^5 + 4x^6 + 4x^7 + 4x^8 + 4x^9 + \cdots with n of each coefficient! This then gives B = ^\infty -x^n\left(^{n-1} x^{k(k+1)/2}\right) = -x-x^2-2x^3-2x^4-2x^5-3x^6-3x^7-3x^8-3x^9-\cdots

ABSTRACT I briefly show how a whole set of formula mimicking the Gauss-Salamin formula can be derived associated with singular values of the complete elliptic integral K(k²), (with elliptic modulus k). MAIN From the Gauss–Salamin formula for π, it can be easily shown that {2}=\left(1,{}\right)K\left({}\right) where K is a complete elliptic integral of the first kind with modulus $k={}$ and agm(x, y) is the arithmetic geometric mean function, which is the shared limit of a∞ or g∞ of the iterative formula a_0={2}(x+y)\\ g_0=\\ a_n={2}(a_{n-1}+g_{n-1})\\ g_n=g_{n-1}} Then if we define F(x)=\left(1,{}\right)K\left({}\right)={4}}{}{}\right)}{K\left(-1}{+1}\right)} we have F\left(2\right)={2} but if we search for more relations we also find that F\left({8}\left(4+3 \right)\right)=\pi\\ F\left(17+12\right)=F\left(\left(1+\right)^4\right)={4}\\ F\left(\left(1+\right)^2\right)={2}\\ F\left(2-\right)=\kappa-{2}i\\ F\left(1+{}\right)=\kappa where κ = 1.6999683849605364588... this gives the identity F\left(2-\right)=F\left(1+{}\right)-iF(2) we see that these identities are in fact independent of π as the values cancel from equation 3. We discover from these various identities about the complete elliptic integral of the first kind, }{}}\right)}{K\left(}-4}{}+4}\right)}=}}{1+{2}{2}}} and \right)}{K\left({}\right)}={4}(2+) and as K\left({}\right)=}{\Gamma(-{4})^2} we have the closed form K\left(3-2\right)=)\pi^{3/2}}{\Gamma\left(-{4}\right)^2}=1.58255... but by letting a=}{8} {}\right)}{K\left(-1}{+1}\right)}=}{1+} It would appear that ({2},{})={\Gamma(1/4)} this directly leads to K(2-3)=)\Gamma(1/4)}{8\Gamma(3/4)}

Idea was that this could be a short opinion piece / overview article (not an Annual Review) with some novel ideas/synthesis. I jotted down some initial thoughts after conversations with David Anderson at Caltech in early 2015. Additional input from another connectomics/cell type person (e.g. Kording, Seung, Ascoli), someone specialising in quantitative approaches to speciation (e.g. Chris Jiggins or David Stern) or one of the people doing DropSeq type work would be interesting.

INVERSE For a 3x3 non-singular matrix A with a determinant |A| defined by A= a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} we can calculate the inverse as A^{-1}={|A|} a_{22} & a_{23} \\ a_{32} & a_{33} & a_{13} & a_{12} \\ a_{33} & a_{32} & a_{12} & a_{13} \\ a_{22} & a_{23} \\ a_{23} & a_{21} \\ a_{33} & a_{31} & a_{11} & a_{13} \\ a_{31} & a_{33} & a_{13} & a_{11} \\ a_{23} & a_{21} \\ a_{21} & a_{22} \\ a_{31} & a_{32} & a_{12} & a_{11} \\ a_{32} & a_{31} & a_{11} & a_{12} \\ a_{21} & a_{22} If we define an operation that is RCRij(A):=Rij, remove the ith column and the jth row of A, then this is expressible as A^{-1}= |R_{11}| & |R_{12}J| & |R_{13}| \\ |R_{21}J| & |R_{22}| & |R_{23}J| \\ |R_{31}| & |R_{32}J| & |R_{33}| Where J= 0 & 1 \\ 1 & 0 Clearly the condition for a right multiplication of the J matrix being i + j = odd. Alternatively |A|(A^{-1})_{ij}=|R_{ij}J^{(i+j-1)}|=|J^{(i+j-1)R_{ij}}| This likely works because for any 2x2 matrix |A|= − |AJ₂|= − |J₂A|=|J₂AJ₂|. This property |A|= − |AJ₂| also appears to hold for a 3x3 matrix. Extrapolating backwards for a two by two matrix we get the correct formula on the proviso we define J₁ ≡ −1. This makes some sense, as for any JnJn = I and JnJnA = A. We can further extrapolate to the inverse of a 1x1 matrix A = A₁₁, taking the R₁₁ element to be the zero matrix, the determinant of this matrix is 1 and the reciprocal of the determinant of A is then just the reciprocal of A₁₁, which again is the inverse of the 1x1 matrix. Proof J₁ = −1: For any Jn, n > 1, |Jn|= − 1 as Jn is defined to be an antidiagonal matrix. |AB|=|A||B|. Therefore |AJn|= − |A|. If one extrapolates to the case n = 1, For the above to remain true, J₁ = −1 OTHER CONCEPT Looking at the same formula we can define four 3x3 matrices, top left, top right, bottom left, bottom right A^{TL}= a_{22} & a_{13} & a_{12} \\ a_{23} & a_{11} & a_{13} \\ a_{21} & a_{12} & a_{11} \\ A^{TR}= a_{23} & a_{12} & a_{13} \\ a_{21} & a_{13} & a_{11} \\ a_{22} & a_{11} & a_{12} \\ A^{BL}= a_{32} & a_{33} & a_{22} \\ a_{33} & a_{31} & a_{23} \\ a_{31} & a_{32} & a_{21} \\ A^{BR}= a_{33} & a_{32} & a_{23} \\ a_{31} & a_{33} & a_{21} \\ a_{32} & a_{31} & a_{22} These matrices are constructed from the rows of the original matrix as such, if R_i is the ith row of the original matrix, and P_n is an operator which cycles that row forward n times we have A^{TL}= R_2^TP_2 & R_1^TP_1 & R_1^TP_2 =A^{r(211)}_{p(212)}\\ A^{TR}= R_2^TP_1 & R_1^TP_2 & R_1^TP_1 =A^{r(211)}_{p(121)}\\ A^{BL}= R_3^TP_2 & R_3^TP_1 & R_2^TP_2 =A^{r(332)}_{p(212)}\\ A^{BR}= R_3^TP_1 & R_3^TP_2 & R_2^TP_1 =A^{r(332)}_{p(121)} These matricies are then such that {|A|}(A^{TL} \circ A^{BR} - A^{TR} \circ A^{BL}) = A^{-1} Where ∘ is the Hadamard product or element-wise product.

ABSTRACT Let P(x) be the generating function for the primes pn, that is P(x)=^\infty p_nx^n We may also define the invert transform as applied the a generating function G(x) as I[G(x)]={1-G(x)}-1 Let I(n)[G(x)] denote a repeated application of the invert transform n times. Then I^{(2)}[G(x)]={1-({1-G(x)}-1)}-1\\ I^{(3)}[G(x)]={1-({1-({1-G(x)}-1)}-1)}-1 We then have I^{(0)}[P(x)]=2x+3x^2+5x^3+7x^4+11x^5+\cdots\\ I^{(1)}[P(x)]=2x+7x^2+25x^3+88x^4+311x^5+\cdots\\ I^{(2)}[P(x)]=2x+11x^2+61x^3+337x^4+1863x^5+\cdots\\ I^{(3)}[P(x)]=2x+15x^2+113x^3+850x^4+6395x^5+\cdots\\ I^{(4)}[P(x)]=2x+19x^2+181x^3+1723x^4+16403x^5+\cdots\\ we can see that if we take all the coefficients of terms xm, m ∈ [1, ∞) we can generate further sequences S_1=2,2,2,2,2,2,2,\cdots\\ S_2=3,7,11,15,19,\cdots\\ S_3=5,25,61,113,181,\cdots\\ S_4=7,88,337,850,1723, \cdots\\ S_5=11,311,1863,6395,1640, \cdots We can then find the sequence function for each of these sequences, for example S_1(n)=2\\ S_2(n)=4n-1\\ S_3(n)=8n^2-4n+1\\ S_4(n)=16n^3-12n^2+5n-2\\ S_5(n)=32n^4-32n^3+18n^2-10n+3\\ These are then polynomials, such that Sk(1)=pk. We note that the constant terms are A030018, the coefficients of the generating function . For a general polynomial, there are relationships we have S_k(n)=^{k} a_i n^{i-1} \\ S_k(1)=^{k} a_i = p_k In general we may note that for Sk(n) a_k = 2^k\\ a_{k-1} = -(k)2^{k-1} \\ a_{k-2} = (3k+k^2)2^{k-3} \\ a_{k-3} = -}{3}(38k+9k^2+k^3) \\ a_{k-4} = }{3}(378k+179k^2+18k^3+k^4) \\ a_{k-5} = -}{15}(9864k+3030k^2+515k^3+30k^4+k^5) \\ a_{k-6} = }{45}(125640k+90634k^2+12915k^3+1165k^4+45k^5+k^6)\\ a_{k-7} = }{315}(1684080k+2003652k^2+463204k^3+40005k^4+2275k^5+63k^6+k^7)\\ a_{k-8} = }{315}(42089040k+50017932k^2+14438676k^3+1728769k^4+101640k^5+4018k^6+84k^7+k^8)\\ a_{k-9} = }{2835}(415900800k+1431527472k^2+492751916k^3+69663132k^4+5253969k^5+225288k^6+6594k^7+108k^8+k^9)\\ However we must shift the series along to deal with the fact that Sk(n) has k terms, and we end up with a_k = 4\cdot2^{k-2}\\ a_{k-1} = -(k-1)2^{k-3} \\ a_{k-2} = (3(k-2)+(k-2)^2)2^{k-5} \\ a_{k-3} = -}{3}(38(k-3)+9(k-3)^2+(k-3)^3)\\ a_{k-4} = }{3}(378+179(k-4)+18(k-4)^2+(k-4)^3) \\ a_{k-5} = -}{15}(9864(k-5)+3030(k-5)^2+515(k-5)^3+30(k-5)^4+(k-5)^5) \\ a_{k-6} = }{45}(125640(k-6)+90634(k-6)^2+12915(k-6)^3+1165(k-6)^4+45(k-6)^5+(k-6)^6) The we can see that the powers of 2 follow a sequence 0, 1, 3, 4, 7, 8, 10, 11, 15, 16, (18?) which is potentially A005187, the number of ones in the binary expansion of 2n, and the denominators of the convergents of $1/$, we will denote this quantity ξ(n), with ξ(0)=0, ξ(1)=1, ⋯. It then appears that for ak − m the coefficent of the highest power of (k − m) is 1, the coefficent of the next highest power is 3m(m + 1)/2 = 3, 9, 18, 30, 35, 63.... We also see the sequence 1, 1, 1, 3, 3, 15, 45, 315, 315, 2835 is A049606, the largest odd divisor of n!. This is very useful. We can attempt to factor n! out of the coefficients. Then, letting χ(n) be A011371, the number of binary digits in n, we have \kappa(n)=}{n!} and a_k = \kappa(0)\\ a_{k-1} = \kappa(1)k \\ a_{k-2} = \kappa(2)(3k+k^2) \\ a_{k-3} = \kappa(3)(38k+9k^2+k^3) \\ a_{k-4} = \kappa(4)(378k+179k^2+18k^3+k^4) \\ a_{k-5} = \kappa(5)(9864k+3030k^2+515k^3+30k^4+k^5) \\ a_{k-6} = \kappa(6)(125640k+90634k^2+12915k^3+1165k^4+45k^5+k^6)\\ a_{k-7} = \kappa(7)(1684080k+2003652k^2+463204k^3+40005k^4+2275k^5+63k^6+k^7)\\ a_{k-8} = \kappa(8)(42089040k+50017932k^2+14438676k^3+1728769k^4+101640k^5+4018k^6+84k^7+k^8)\\ a_{k-9} = \kappa(9)(415900800k+1431527472k^2+492751916k^3+69663132k^4+5253969k^5+225288k^6+6594k^7+108k^8+k^9)\\ a_{k-10} = \kappa(10)(501863040k+39753346896k^2+17788750740k^3+2992825520k^4+261174375k^5+13782153k^6+451710k^7+10230k^8+135k^9+k^{10})\\ a_{k-11} = \kappa(11)(228247891200k+1120677132960k^2+670268327256k^3+132747091620k^4+13570264070k^5+820667925k^6+32340693k^7+838530k^8+15180k^9+165k^{10}+k^{11})\\ a_{k-12} = \kappa(12)(11086611782400k + 34639931748960k^2+25815979252008k^3+6164459073916k^4+720004626990k^5+50304599015k^6+2260051794k^7+69522783k^8+1464210k^9+21725k^{10}+198k^{11}+k^{12}) \\ a_{k-13} \to () Now we focus on the polynomial aspect in k. We may define a function πn(k) such that a_{k-m}= \kappa(m)\pi_m(k) then we have \pi_0(k)=1\\ \pi_1(k)=k\\ \pi_2(k)=3k+k^2\\ \pi_3(k)=38k+9k^2+k^3\\ \pi_4(k)=378k+179k^2+18k^3+k^4 From the above we can see that \pi_m(k) = k^m + {2}k^{m-1} + {24}k^{m-2} + {16}k^{m-3} + {5760}k^{m-4} \\ + {3840}k^{m-5} + we can then expect a reduction \pi_m(k) = k^m + ^{m-1}c_i\left[^{i} (m+j)\right]\sigma_i(m)k^{m-i} where \sigma_1=3\\ \sigma_2=(125+27m)\\ \sigma_3=(118+125m+9m^2)\\ However after shifting to adjust the sequences we have \pi_m(k) = k^m + {2}k^{m-1} + {24}k^{m-2} + {16}k^{m-3} + {5760}k^{m-4} \\ + {3840}k^{m-5} + {2903040}k^{m-6} + \frac{}{1935360} which can then be re-written as \pi_m(k) = k^m + {2}k^{m-1} + {24}k^{m-2} + {16}k^{m-3} + {5760}k^{m-4} \\ + {3840}k^{m-5} + {2903040}k^{m-6} Now the denominators may be explained by the sequence A053657 d_n=(n)=\prod_p p^{^\infty \lfloor {(p-1)p^k} \rfloor } = 1, 2, 24, 48, 5760, 11520, 2903040, 5806080... Then which can then be re-written as \pi_m(k) = {d_1} + {d_2}k^{m-1} + {d_3}k^{m-2} + {d_4}k^{m-3} + {d_5}k^{m-4} \\ + {d_6}k^{m-5} + {d_7}k^{m-6} we can see the subsequence 71, 213, 19170, 19170, 1811565 is 1, 3, 270, 270, 25515 when divided by 71. The number of 3’s in the lead coeficcient are 1, 3, 3, 5, 6, 8 which could be a number of sequences. The number of 3’s in the second coefficients are 0, 1, 3, 3, 6 which could also be many sequences. Then we have \pi_m(k) = ^{m-1}^{i} (m-j)\right]}{d_{i+1}}\sigma_i(m)k^{m-i} and we focus on the σi(m) polynomials \sigma_0(m)=3 = 3(1)\\ \sigma_1(m)=27m+71 \\ \sigma_2(m)=27m^2+213m-528 = 3 (9 m^2+71 m-176)\\ \sigma_3(m)=1215m^3+19170 m^2-69835 m+191522\\ \sigma_4(m)=729m^4+19170m^3-66945m^2+199686 m-1863360 = 3 (243 m^4+6390 m^3-22315 m^2+66562 m-621120)\\ \sigma_5(m)=45927 m^5+1811565 m^4-3671325 m^3-20584781 m^2-243156858 m-181415824\\ \sigma_6(a)=3(6561 a^6+362313 a^5+273105 a^4-22127777 a^3+15860502 a^2-1815371056 a+12146754816)\\ \sigma_7(a)=885735 a^7+65216340 a^6+316232910 a^5-8968949640 a^4+26098075255 a^3-1410488924700 a^2+16110491218964 a-46896294366576\\ \sigma_8(a)=3(98415 a^8+9316620 a^7+95579190 a^6-2167195968 a^5+5460923335 a^4-506938037980 a^3+7682473899284 a^2-34005560902256 a+42398207462400)\\ \sigma_9(a)= We find that the sequence 0, 3, 27, 27, 1215, 729, 45927, 19683, 885735, 295245, 5845851, 1594323, ... is given by C_m=\left((n)}{n!} \right)3^{m-1}-2(-1)^{m-1}+(-2)^{m-1})}{2}\Gamma where \left((n)}{n!} \right)= 1, 1, 1, 1, 1, 3, 1, 9, 9, 15, 3, 9, 3, 945, 135, 27, 27, 405, 45, 8505, 1701, 66825, 6075, 18225, 6075, 995085, 76545,\\ 3^{m-1}= 1,3,9,27...\\ -2(-1)^{m-1}+(-2)^{m-1})}{2}= 0, 1, 3, 1, 15, 1, 63, 1, 255, 1, 1023, 1, 4095, 1, 16383, 1 \\ \Gamma= 1,1,1,1,1,1,1,1,{17},1,{31},1,1,1,{5461},1,{257}... We can see this is related by {N[tan(x)]}=1,1,1,{17},{31},1,{5461},{257},{73},{1271} Where N[f(x)] is the numerator of a Taylor expanded f. However there is one later coefficient which doesn’t seem to agree. 4097 in the original sequence has in this cot/tan expansion 241 which is 17 times too small. Odd that 17 is the first coefficient, also later in the sequence is the term 32505887 which doesn’t agree with the cot/tan term 61681. However, the former divied by the latter is 17 ⋅ 31, i.e. the product of the first two terms.

ABSTRACT This document explains a method of creating an integer sequence by a counting rule. The integer sequence made relates to the place values of the digits in numbers created by the counting rule. Such rules create the numbers bases b, the factorial base numbers, the Catalan numbers, the Motzkin numbers, and more. MAIN When counting regularly in a given base b, the ’rule’ of counting is that we increment the right most digit by one until it exceeds b − 1. At this point we carry a 1 to the next digit, (apply the rule again further up the number if necessary). This document goes beyond the concept of a constant b, and extends to arbitrary rules for counting. This concept has already been seen in the factorial counting system, where for the kth digit in a number (indexed from 1), that digit may not exceed k. Such a rule leads to the values of each digit or place value, being the factorial numbers, as opposed to the regular powers bk. For the base b numbers we can write the ’rule’ for when to carry a 1, (d_k)\;\; d_k > b-1 for the factorial base we can write (d_k)\;\; d_k > k where ’carry’ is the operation, “set d to zero and let dk + 1 = dk + 1 + 1”. The table below displays some examples for counting in the factorial base & \\ \\ 0 & 0 \\ 1 & 1 \\ 10 & 2 \\ 11 & 3 \\ 20 & 4 \\ 21 & 5 \\ 100 & 6 \\ \cdots \\ 321 & 23 \\ 1000 & 24 \\ \cdots \\ 4321 & 119 \\ 10000 & 120 In order for the left notation generated by the rule to equal the decimal notation on the right, the digits must be worth 1, 2, 6, 24, ..., the factorial numbers. The problem with general rules is that a countably infinite number of unique symbols are required to represent arbitrarily large numbers. Thus they do not make ’nice’ natural counting systems from a social perspective. However, from a mathematical perspective, the numbers that equal 1, 10, 100, 1000, ⋯ and so on have combinatorial interpretations. CATALAN NUMBERS We can generate a simple rule where the digits dk of a number written in the ’Catalan Base’ must equal the Catalan numbers A000108(k). The rule is that a digit is carried when it exceeds the digit to its left plus one. We can again write more concisely (d_k)\;\; d_k > d_{k+1}+1 & \\ \\ 0 & 0 \\ 1 & 1 \\ 10 & 2 \\ 11 & 3 \\ 12 & 4 \\ 100 & 5 \\ 101 & 6 \\ 110 & 7 \\ 111 & 8 \\ 112 & 9 \\ 120 & 10 \\ 121 & 11 \\ 122 & 12 \\ 123 & 13 \\ 1000 & 14 We see the individual digits must take the values 1, 2, 5, 14, ..., which make the Catalan numbers. However, unlike the factorial base numbers we easily lose the property of the notation that the sum of the digits weighted by their place values equals the decimal value to the right. For example 120 → 10, but 5 + 2 + 2 ≠ 10. Due to this it is not strictly speaking correct to call the numbers 1, 10, 100, 1000 place values. However, when the counting system arrives at these values, it denotes the end of a cycle of some kind through permutations of the lower digits. MOTZKIN NUMBERS If we add an extra condition, that we carry if current digit is greater than the next digit OR the next next digit plus 1, then we arrive at the Motzkin numbers, A001006. (d_k)\;\; d_k > d_{k+1}+1 \; \; d_k > d_{k+2} + 1 there is clearly a combinatorial interpretation of the way the numbers are being counted. ’AND’ MOTZKIN NUMBERS What happens if we change the OR for an AND in the above example? (d_k)\;\; d_k > d_{k+1}+1 \; \; d_k > d_{k+2} + 1 This produces sequence A275605, which was not previously in the OEIS. Does this sequence have a nice combinatoric interpretation? A254439 This methodology along with some experimentation produced the terms of a sequence with the “more” flag, on OEIS, we have the rule (d_k)\;\; d_k > 3d_{k+1}+1 which produces place values of 1, 2, 7, 47, 682, 23132, 1913821, 397731998, ⋯. It is possible to explain these terms using a formal but increasingly intractable ’chain of sums’ 7 = ^2 (3i_1-1)\\ 47 = ^2 ^{3i_1-1} (3i_2-1) \\ 682 = ^2 ^{3i_1-1} ^{3i_2-1} (3i_3-1) \\ \cdots \\ a(n) = ^2 \cdots ^{3i_{n-1}-1}}_{} (3i_n-1) TABULATION OF RULES AND SEQUENCES The following table is not rigorously proved! I have tabulated some results I have seen so far, some have no entry in the OEIS. & \\ d_k > b-1 & \\ d_k > k & \\ d_k > d_{k+1}+1 & \\ d_k > d_{k+1}+1 \; \; d_k > d_{k+2} + 1 & \\ d_k > d_{k+1}+1 \; \; d_k > d_{k+2} + 1 & \\ d_k > d_{k+1}+2 \; \; d_k > d_{k+2}+2 & A063020\\ d_k > d_{k+2}+1 & A005817 \\ d_k > 2d_{k+1}+1 & A002449 \\ d_k > d_{k+2}+2 & 1, 3, 9, 36, 144, 660, 3025, 15015, 74529, 389844, 2039184, 11069856, 60093504 \\ d_k^2 > d_{k+1}+1 & A000079 & \\ d_k^2 > d_{k+1}+d_{k}+1 & A001519 \\ d_k^2 > d_{k+1}+d_{k}+2 & A000244 & \\ d_k^2 + d_{k+1}^2 > d_{k+2}^2 + 1 & A000045 & \\ d_k^2 > d_{k+1}^2 + d_{k+2}^2 + 2 & 1, 2, 4, 9, 21, 52, 133, 353, 963, 2705, 7812, 23289, 72057, 233771, 807131, 3028323, 12676939, 60959699, 345934925 \\ d_k^2 > 2*d_{k+1}*d_{k+2} + 2 & 1, 2, 4, 9, 21, 51, 127, 323, 836, 2202, 5910, 16222, 45821, 134565, 417568, 1403052, 5279866, 23142475, 122222895 \\ d_k > d_{k+1}*d_{k+2} + 1 & 1, 2, 4, 9, 22, 64, 288, 5758, 2715615 \\ d_k > d_{k+1} + d_{k+2} + 1 & 1, 2, 5, 17, 74, 425, 3229, 33220, 470345, 9341720 \\ d_k > d_{k+1} + 1 & 1, 2, 5, 16, 99, 8764, 464897407 \\ d_k > d_{k+3} + 1 & 1, 2, 4, 8, 20, 50, 125, 350, 980, 2744, 8232, 24696, 74088, 232848, 731808, 2299968, 7474896, 24293412, 78953589\\ d_k > d_{k+1} + 1 \;\; d_k > d_{k+2} + 1 \;\; d_k > d_{k+3} + 1& A004148\\ d_k > d_{k+1} + k & A101482 \\ d_k + d_{k+1} > d_{k+2} + 1 & 1, 2, 3, 6, 10, 358056266, 1431798090, 3904618790 \\ d_k > d_{k+2} + d_{k+3} + 1 & 1, 2, 4, 10, 31, 106, 444, 2153, 12200, 82857, 659178, 6292825, 72012699 \\ d_k > d_{k+2} + d_{k+3} - d_{k+1} + 1 & 1, 2, 3, 6, 12, 25, 54, 118, 271, 624, 1487, 3572, 8795, 21874, 55340, 141593, 366858, 960966, 2542355, 6793664, 18307745, 49774625 \\ d_k > ^2}{2} + 2 & 1, 3, 11, 51, 441, 53371 \\ d_k > k^2 & A101686 \\ d_k > k^2 -k & A130032 \\ d_k > e^k & 1, 3, 24, 504, 27720, 4130280, \\ d_k > log(k) & 1, 2, 4, 8, 16, 32, 96, 288, 864, 2592, 7776, 23328, 69984, 209952, 629856, 1889568, 5668704, 17006112 \\ d_k > d_{k+1} + log(k) & 1, 4, 14, 48, 165, 572, 5070, 38050, 267252, 1813998 \\ d_k > d_{k+1} + k^2 & 1, 2, 20, 635, 43095, 5075200, 919589971 \\ d_k > |d_{k+1}-d_{k+2}| & 1, 2, 5, 13, 36, 104, 307, 930, 2855, 8883, 27950, 88668, 283555, 912647, 2954111, 9611223, 31408216 \\ d_k > + 1 & 1, 3, 9, 27, 108, 432, 1728, 6912, 27648, 138240, 691200, 3456000, 17280000 \\ d_k > & 1, 2, 4, 8, 24, 72, 216, 648, 1944, 7776, 31104, 124416, 497664, 1990656, 7962624 \\ d_k > (d_{k+1},d_{k+2}) & 1, 2, 5, 14, 43, 137, 446, 1477, 4942, 16656, 56439, 191995, 655119, 2240706, 7678343, 26351967 \\ d_k > (d_{k+1},3) & A003946 \\ d_k > (d_{k+1} \; \; 2) + 1& A000129 \\ d_k > (d_{k+1}+1 \; \; 2) + 1 & A001906 \\ d_k > (d_{k+1} \; \; 3) + 1& A202207 \\ d_k > (d_{k+1}+1 \; \; 3) + 1& A084084 \\ d_k > (d_{k+1}+2 \; \; 3) + 1& A052529 \\ d_k > (d_{k+2} \; \; 2) + 1& A089928 \\ d_k > (d_{k+1} \; \; 2)(d_{k+2} \; \; 2) + 1& A008998 \\ d_k > )}{k} + 1& A084215 \\ d_k > } + 1 & A166516 \\ d_k > } + 1 & A001519 \\ d_k > \sin(k\pi/2)+1 & A026532 \\ d_k > 2\sqrt(d_{k+1})+1 & A124292 \\ d_k > {2}}+1 & A124302 \\ d_k > {3}}+1 & A001519 \\ d_k > 2}}+1 & A006012 \\ d_k > d_{k+d_{k+1}+1}+1 & 1, 2, 4, 9, 21, 51, 128, 331, 879, 2394, 6685, 19123, 56006, 167878, 514941, 1616006, 5186700 \\ d_k > d_{k+d_{k}+1}+1 & 1, 2, 4, 8, 20, 50, 125, 325, 890, 2440, 6744, 18996, 54090, 155290, 448586\\ d_k > d_{k+d_{k}}+1& 1, 2, 4, 10, 25, 65, 178, 492, 1386, 3954, 11426, 33275, 97633, 288384, 856355, 2555366, 7656046, 23022604 \\ d_k > } + 1 & A007052 \\ d_k > } + 1 & A001835 \\ d_k > } + 1 & A000302 & \\ d_k > } + 1 & A052913 EXTENSIONS We can implement other variables into the ’rule’. For example, the previous place value can feature. If we start a variable p = 0, at the start of counting, after the next term is found we can increment p, then p may directly appear in the rule to generate even more complicated rules. Finding a known sequence with these rules is very rare so far. Some examples are & \\ d_{k} > k -{2} + 1 & 1, 3, 7, 25, 43, 139, 235, 835, 1435, 5755, 10075, 45355, 80635, 403195, 725755, 3991675, 7257595, 43545595, 79833595 \\ d_{k} > k-p + 1 & 1,2,3,4,5,6,7,8,9 \\ d_{k} > k-{\pi} & 1, 2, 3, 7, 11, 29, 125, 221, 821, 5141, 9461, 44741, 367301, 689861, 3955781, 40243781 \\ d_{k} > d_{k+1} + 1 + p & 1, 3, 21, 271, 5246, 136552, 4489516, 178921726 \\ d_{k} > k - {e} & 1, 2, 3, 4, 8, 26, 44, 140, 740, 1340, 5660, 9980, 45260, 367820, 690380, 3956300, 40244300, 76532300 \\ REFERENCES N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org