A054890 Layer counting sequence for hyperbolic tessellation by regular heptagons of angle Pi/3.

1, 7, 42, 245, 1428, 8323, 48510, 282737, 1647912, 9604735, 55980498, 326278253, 1901689020, 11083855867, 64601446182, 376524821225, 2194547481168, 12790760065783, 74550012913530, 434509317415397

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

• Michael De Vlieger, Table of n, a(n) for n = 1..1307
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• 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 (6,-1).

Crossrefs

Cf. A001109, A054886, A054887, A054888, A054889.

Programs

Magma

[n eq 1 select 1 else 7*Evaluate(ChebyshevSecond(n-1), 3): n in [1..40]]; // G. C. Greubel, Feb 08 2023

Mathematica

Rest@CoefficientList[Series[x*(1+x+x^2)/(1-6*x+x^2), {x,0,30}], x] (* Michael De Vlieger, Dec 29 2020 *)
LinearRecurrence[{6,-1},{1,7,42},20] (* Harvey P. Dale, Jun 06 2021 *)

SageMath

[7*chebyshev_U(n-2, 3) + int(n==1) for n in range(1,41)] # G. C. Greubel, Feb 08 2023

Formula

a(n) = 7*A001109(n-1) + [n=1].

G.f.: x*(1+x+x^2)/(1-6*x+x^2).

a(n) = A001109(n) + A001109(n-1) + A001109(n-2), n>1. - Ralf Stephan, Apr 26 2003