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%I #54 Feb 09 2023 02:09:33
%S 1,5,15,40,105,275,720,1885,4935,12920,33825,88555,231840,606965,
%T 1589055,4160200,10891545,28514435,74651760,195440845,511670775,
%U 1339571480,3507043665,9181559515,24037634880,62931345125
%N Layer counting sequence for hyperbolic tessellation by regular pentagons of angle Pi/2.
%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.
%H Reinhard Zumkeller, <a href="/A054888/b054888.txt">Table of n, a(n) for n = 0..999</a> (indices corrected to start at zero by _Sidney Cadot_, Jan 07 2022)
%H Paolo Dominici, <a href="/layers.html">Illustration</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_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F a(n) = 5*A001906(n) + [n=0].
%F G.f.: (1+x)^2/(1-3*x+x^2).
%F G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^2 * x^n/n ). - _Paul D. Hanna_, Feb 21 2012
%F a(n) = A001906(n-1) + 2*A001906(n) + A001906(n+1). - _R. J. Mathar_, Nov 28 2011
%F a(n) = A203976(A004277(n-1)). - _Reinhard Zumkeller_, Jan 11 2012
%F a(n) = 5*A000045(2*n) for n >= 1. - _Robert Israel_, Jun 01 2015
%t LinearRecurrence[{3,-1},{1,5,15},30] (* _Harvey P. Dale_, Jan 15 2023 *)
%t Join[{1}, 5*Fibonacci[2*Range[40]]] (* _G. C. Greubel_, Feb 08 2023 *)
%o (Haskell)
%o a054888 n = a054888_list !! (n-1)
%o a054888_list = 1 : zipWith (+) (tail a002878_list) a002878_list
%o -- _Reinhard Zumkeller_, Jan 11 2012
%o (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 */
%o (Magma) [n eq 0 select 1 else 5*Fibonacci(2*n): n in [0..40]]; // _G. C. Greubel_, Feb 08 2023
%o (SageMath) [5*fibonacci(2*n) + int(n==0) for n in range (41)] # _G. C. Greubel_, Feb 08 2023
%Y Cf. A000045, A001906, A002878, A004277, A203976.
%Y Cf. A054886, A054887, A054889, A054890.
%K nonn,easy
%O 0,2
%A Paolo Dominici (pl.dm(AT)libero.it), May 23 2000
%E Offset changed to 0 by _N. J. A. Sloane_, Jan 03 2022 at the suggestion of _Michel Marcus_
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