A054889 Layer counting sequence for hyperbolic tessellation by regular pentagons of angle 2*Pi/5.

1, 5, 20, 70, 245, 860, 3015, 10570, 37060, 129935, 455560, 1597225, 5599980, 19633910, 68837825, 241350100, 846189875, 2966799290, 10401800220, 36469419475, 127864266640, 448300820765, 1571773187140, 5510743762630

Offset

1,2

Comments

The layer sequence is the sequence of the cardinalities of the layers accumulating around a (finite-sided) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888.

Links

Georg Fischer, Table of n, a(n) for n = 1..500

Index entries for Coordination Sequences [A layer sequence is a kind of coordination sequence. - N. J. A. Sloane, Nov 20 2022]

Index entries for linear recurrences with constant coefficients, signature (3,1,3,-1).

Formula

G.f.: x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4).

Mathematica

LinearRecurrence[{3, 1, 3, -1}, {1, 5, 20, 70, 245}, 40] (* Georg Fischer, Apr 13 2020 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4) )); // G. C. Greubel, Feb 08 2023
(Sage)
def A054889_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x+4*x^2+2*x^3+x^4)/(1-3*x-x^2-3*x^3+x^4) ).list()
a=A054889_list(40); a[1:] # G. C. Greubel, Feb 08 2023

Crossrefs

Cf. A054886, A054887, A054888, A054890.

Sequence in context: A304011 A243869 A154638 * A056384 A369818 A036683

Adjacent sequences: A054886 A054887 A054888 * A054890 A054891 A054892

Keyword

nonn,easy

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Extensions

a(21) inserted by Georg Fischer, Apr 13 2020

Status

approved