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

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A054886, A054887, A054888, A054890.

Programs

Magma

R:=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

Mathematica

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

Sage

def A054889_list(prec):
    P. = 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

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).

Extensions

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