|
|
A369472
|
|
Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.
|
|
9
|
|
|
1, 1, 2, 4, 9, 22, 52, 140, 340, 969, 2394, 7084, 17710, 53820, 135720, 420732, 1068012, 3362260, 8579560, 27343888, 70068713, 225568798, 580034052, 1882933364, 4855986044, 15875338990, 41043559340, 134993766600
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link.
|
|
LINKS
|
|
|
FORMULA
|
For n even, a(n) = C(2n,n/2)/(3n/2+1).
For n odd, a(n) = 4*C(2n-1,(n-1)/2)/(3n+1).
a(n+2)/a(n) ~ 256/27. a(2m+1)/a(2m) ~ 32/9; a(2m)/a(2m-1) ~ 8/3.
G.f.: G(z^2)+z*G(z^2)^2, where G(z)=1+z*G(z)^4, the generating function for A002293.
|
|
MATHEMATICA
|
p=5; Table[If[EvenQ[n], Binomial[(p-1)n/2, n/2]/((p-2)n/2+1), If[OddQ[p], (p-1)Binomial[(p-1)n/2-1, (n-1)/2]/((p-2)n+1), p Binomial[(p-1)n/2-1/2, (n-1)/2]/((p-2)n+2)]], {n, 35}]
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|