A054887 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3, Pi/5, Pi/7).

1, 3, 6, 11, 20, 36, 64, 113, 200, 354, 626, 1107, 1958, 3464, 6128, 10839, 19172, 33913, 59988, 106111, 187696, 332009, 587280, 1038820, 1837534, 3250353, 5749442, 10169998, 17989372, 31820803, 56286764, 99563792, 176115092

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

G. C. Greubel, Table of n, a(n) for n = 1..1000

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 (0,0,2,2,4,3,4,2,2,0,0,-1).

Formula

G.f.: x*(1+x)*(1-x^3)*(1-x^5)*(1-x^7)/(1-2*x+x^4+x^6-x^10-x^12+2*x^15-x^16).

Mathematica

LinearRecurrence[{0, 0, 2, 2, 4, 3, 4, 2, 2, 0, 0, -1}, {1, 3, 6, 11, 20, 36, 64, 113, 200, 354, 626, 1107, 1958}, 41] (* G. C. Greubel, Feb 07 2023 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+x)*(1-x^3)*(1-x^5)*(1-x^7)/(1-2*x+x^4+x^6-x^10-x^12+2*x^15-x^16) )); // G. C. Greubel, Feb 07 2023
(SageMath)
def A054887_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x)*(1-x^3)*(1-x^5)*(1-x^7)/(1-2*x+x^4+x^6-x^10-x^12+2*x^15-x^16) ).list()
a=A054887_list(40); a[1:] # G. C. Greubel, Feb 07 2023
(PARI) Vec(x*(1+x+x^2)*(1+x+x^2+x^3+x^4)*(1+x+x^2+x^3+x^4+x^5+x^6)/(1-2*x^3-2*x^4-4*x^5-3*x^6-4*x^7-2*x^8-2*x^9+x^12)+O(x^99)) \\ Charles R Greathouse IV, May 25 2026

Crossrefs

Cf. A054888.

Sequence in context: A055417 A018918 A077855 * A019302 A119861 A255061

Adjacent sequences: A054884 A054885 A054886 * A054888 A054889 A054890

Keyword

nonn,easy

Author

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

Status

approved