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Layer counting sequence for hyperbolic tessellation by regular pentagons of angle 2*Pi/5.
3

%I #15 Apr 19 2023 02:16:11

%S 1,5,20,70,245,860,3015,10570,37060,129935,455560,1597225,5599980,

%T 19633910,68837825,241350100,846189875,2966799290,10401800220,

%U 36469419475,127864266640,448300820765,1571773187140,5510743762630

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

%C 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.

%H Georg Fischer, <a href="/A054889/b054889.txt">Table of n, a(n) for n = 1..500</a>

%H <a href="/index/Con#coordseqs">Index entries for Coordination Sequences</a> [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,3,-1).

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

%t LinearRecurrence[{3,1,3,-1},{1,5,20,70,245},40] (* _Georg Fischer_, Apr 13 2020 *)

%o (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

%o (Sage)

%o def A054889_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o 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()

%o a=A054889_list(40); a[1:] # _G. C. Greubel_, Feb 08 2023

%Y Cf. A054886, A054887, A054888, A054890.

%K nonn,easy

%O 1,2

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

%E a(21) inserted by _Georg Fischer_, Apr 13 2020