# OEIS/coors

## Generating functions for coordination sequences of uniform tilings

For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a **tab-separated file** with the result.

As an example we examine the two lines (coseq.s) for **Gal.2.13** in this file:

2-Uniform Tiling 13 of 20 Standard notation: [3^6; 3^2.4.12] A301692 Gal.2.13.1 O.g.f.: (-x^13-2*x^12+3*x^10+5*x^9+8*x^8+11*x^7+13*x^6+14*x^5+12*x^4+9*x^3+7*x^2+4*x+1) / (x^10-2*x^5+1) Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1 a14>s11 Denom. factorized: (x^5-1)^2 A301690 Gal.2.13.2 O.g.f.: (x^10+6*x^9+6*x^8+9*x^7+12*x^6+16*x^5+12*x^4+9*x^3+6*x^2+6*x+1) / (x^10-2*x^5+1) Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1 s11=s11 Denom. factorized: (x^5-1)^2

The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:

- The coefficient lists for the denominators are always, those for the numerators are often
**symmetrical**(the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical. - All coefficient lists start and end with coefficent 1.
- If both the numerator and the denominator coefficient lists are symmetrical, the degrees are both odd (the leading numbers give the occurrences).

### Factors of the denominator polynomials

For the denominators of the g.f.s several factorization mathods (Mathematica *FactorList* and *Simplify*, Maple *factor*) were tried with different results. In the end, a Maple procedure of Robert Israel was used:

myfactor:= proc(p) local X, k, P, T, q; P:= p; T:= 1; X:= indets(p)[1]; k:= degree(P,X); while k > 0 do if rem(P, X^k-1, X, 'q') = 0 then P:= q; T:= T*(X^k-1) else k:= k-1 fi od; T * factor(P) end proc;

With this factorization:

- Most of the factors have the form
*x^k-1*. - There are no coefficients > 1.
- Some of the polynomials could be further reduced by a substitutions
*x^m -> y*. - After such substitutions, the factors have the form
*sum(k=0..n: (+-1)^k * x^k)*, except for one polynomial of degree 8 and two polynomials of degree 12.

There are the following 55 different factors in the denominators (with their number of occurrences prefixed):

892 x+1 2091 x-1 1613 x^2+1 946 x^2+x+1 233 x^2-1 1188 x^2-x+1 981 x^3-1 975 x^4+1 621 x^4+x^3+x^2+x+1 205 x^4-1 884 x^4-x^2+1 783 x^4-x^3+x^2-x+1 1260 x^5-1 445 x^6+x^3+1 205 x^6+x^5+x^4+x^3+x^2+x+1 360 x^6-1 57 x^6-x^3+1 122 x^6-x^5+x^4-x^3+x^2-x+1 1255 x^7-1 130 x^8+1 468 x^8-1 20 x^8-x^4+1 38 x^8-x^6+x^4-x^2+1 124 x^8-x^7+x^5-x^4+x^3-x+1 1144 x^9-1 145 x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 628 x^10-1 6 x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1 1492 x^11-1 6 x^12+x^11-x^9-x^8+x^6-x^4-x^3+x+1 182 x^12-1 2 x^12-x^10+x^8-x^6+x^4-x^2+1 5 x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1 12 x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1 5 x^12-x^6+1 452 x^13-1 117 x^14-1 198 x^15-1 6 x^16+1 29 x^16-1 89 x^17-1 6 x^18+x^9+1 57 x^18-1 59 x^19-1 10 x^20-1 20 x^21-1 6 x^23-1 3 x^24-1 18 x^25-1 6 x^27-1 32 x^28-1 6 x^30-1 6 x^35-1 6 x^36-1 6 x^40-1