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

Author

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

Keywords

• nonn

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).

Crossrefs

Cf. A054888.

Programs

Magma

R:=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

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 *)

Sage

def A054887_list(prec):
    P. = 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

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).