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 seuences were guessed by Maple's gfun:guessgf from 128 initial terms.

The resulting rational g.f.s show - of course ? - rather obvious symmetrical properties.

Denominator polynomials

When the denomiator polynomials are factored, there remain only the following 20, essentially different factors:

  6536 x-1
  5979 x+1

  2850 x^2-x+1
  4605 x^2+x+1

  1306 x^4-x^3+x^2-x+1
  2303 x^4+x^3+x^2+x+1

   305 x^6-x^5+x^4-x^3+x^2-x+1
  1474 x^6+x^5+x^4+x^3+x^2+x+1

   328 x^8-x^7+x^5-x^4+x^3-x+1
     6 x^8+x^7-x^5-x^4-x^3+x+1

     6 x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
  1492 x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1

    32 x^12-x^11     +x^9-x^8    +x^6    -x^4+x^3-x+1
     6 x^12+x^11     -x^9-x^8    +x^6    -x^4-x^3+x+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
   452 x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1

    89 x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1

    59 x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1

     6 x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1

     6 x^24-x^23+x^19-x^18+x^17-x^16+x^14-x^13+x^12-x^11+x^10-x^8+x^7-x^6+x^5-x+1

Sometimes a mapping of x^n -> x was applied in these factors (n >= 2).