A054888 Layer counting sequence for hyperbolic tessellation by regular pentagons of angle Pi/2.

1, 5, 15, 40, 105, 275, 720, 1885, 4935, 12920, 33825, 88555, 231840, 606965, 1589055, 4160200, 10891545, 28514435, 74651760, 195440845, 511670775, 1339571480, 3507043665, 9181559515, 24037634880, 62931345125

Offset: 0

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.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..999 (indices corrected to start at zero by _Sidney Cadot_, Jan 07 2022)
• Paolo Dominici, Illustration
• 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).

Crossrefs

Cf. A000045, A001906, A002878, A004277, A203976.

Cf. A054886, A054887, A054889, A054890.

Programs

Haskell

a054888 n = a054888_list !! (n-1)
a054888_list = 1 : zipWith (+) (tail a002878_list) a002878_list
-- Reinhard Zumkeller, Jan 11 2012

Magma

[n eq 0 select 1 else 5*Fibonacci(2*n): n in [0..40]]; // G. C. Greubel, Feb 08 2023

Mathematica

LinearRecurrence[{3,-1},{1,5,15},30] (* Harvey P. Dale, Jan 15 2023 *)
Join[{1}, 5*Fibonacci[2*Range[40]]] (* G. C. Greubel, Feb 08 2023 *)

PARI

{a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^2*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna, Feb 21 2012 */

SageMath

[5*fibonacci(2*n) + int(n==0) for n in range (41)] # G. C. Greubel, Feb 08 2023

Formula

a(n) = 5*A001906(n) + [n=0].

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

G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^2 * x^n/n ). - Paul D. Hanna, Feb 21 2012

a(n) = A001906(n-1) + 2*A001906(n) + A001906(n+1). - R. J. Mathar, Nov 28 2011

a(n) = A203976(A004277(n-1)). - Reinhard Zumkeller, Jan 11 2012

a(n) = 5*A000045(2*n) for n >= 1. - Robert Israel, Jun 01 2015

a(n) = A002878(n-1)+A002878(n). - R. J. Mathar, Jul 09 2024

Extensions

Offset changed to 0 by N. J. A. Sloane, Jan 03 2022 at the suggestion of Michel Marcus