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%I #18 Feb 09 2023 02:38:41
%S 1,7,42,245,1428,8323,48510,282737,1647912,9604735,55980498,326278253,
%T 1901689020,11083855867,64601446182,376524821225,2194547481168,
%U 12790760065783,74550012913530,434509317415397
%N Layer counting sequence for hyperbolic tessellation by regular heptagons of angle Pi/3.
%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; see the illustration accompanying A054888.
%H Michael De Vlieger, <a href="/A054890/b054890.txt">Table of n, a(n) for n = 1..1307</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%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 (6,-1).
%F a(n) = 7*A001109(n-1) + [n=1].
%F G.f.: x*(1+x+x^2)/(1-6*x+x^2).
%F a(n) = A001109(n) + A001109(n-1) + A001109(n-2), n>1. - _Ralf Stephan_, Apr 26 2003
%t Rest@CoefficientList[Series[x*(1+x+x^2)/(1-6*x+x^2), {x,0,30}], x] (* _Michael De Vlieger_, Dec 29 2020 *)
%t LinearRecurrence[{6,-1},{1,7,42},20] (* _Harvey P. Dale_, Jun 06 2021 *)
%o (Magma) [n eq 1 select 1 else 7*Evaluate(ChebyshevSecond(n-1), 3): n in [1..40]]; // _G. C. Greubel_, Feb 08 2023
%o (SageMath) [7*chebyshev_U(n-2, 3) + int(n==1) for n in range(1,41)] # _G. C. Greubel_, Feb 08 2023
%Y Cf. A001109, A054886, A054887, A054888, A054889.
%K nonn
%O 1,2
%A Paolo Dominici (pl.dm(AT)libero.it), May 23 2000
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