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%I #71 Sep 19 2023 15:33:04
%S 1,-4,15,-56,209,-780,2911,-10864,40545,-151316,564719,-2107560,
%T 7865521,-29354524,109552575,-408855776,1525870529,-5694626340,
%U 21252634831,-79315912984,296011017105,-1104728155436,4122901604639,-15386878263120,57424611447841
%N a(0) = 1, a(1) = -4, a(n) = -4*a(n-1) - a(n-2) for n > 1.
%C Pisano period lengths: 1, 2, 3, 4, 6, 6, 8, 4, 9, 6, 5, 12, 12, 8, 6, 8, 9, 18, 10, 12, ... - _R. J. Mathar_, Aug 10 2012
%C In engineering literature, these numbers are known as Clapeyron numbers, or Clapeyron's numbers, or Clapeyronian numbers, on account of their appearance in Benoît Clapeyron's influential study (1857) of the bending forces imposed upon multiple supports of a horizontal beam. - _John Blythe Dobson_, Mar 12 2014
%D Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967, pp. 35-46.
%H Vincenzo Librandi, <a href="/A125905/b125905.txt">Table of n, a(n) for n = 0..1000</a>
%H [Benoît] Clapeyron, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k30026/f1081.image">Calcul d'une poutre élastique reposant librement sur des appuis inégalement espacés</a>, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 45 (1857), 1076-1080.
%H Felix Flicker, <a href="https://arxiv.org/abs/1707.09371">Time quasilattices in dissipative dynamical systems</a>, arXiv:1707.09371 [nlin.CD], 2017. Also <a href="http://doi.org/10.21468/SciPostPhys.5.1.001">SciPost</a> Phys. 5, 001 (2018).
%H Pavel Galashin, Alexander Postnikov, and Lauren Williams, <a href="https://arxiv.org/abs/1909.05435">Higher secondary polytopes and regular plabic graphs</a>, arXiv:1909.05435 [math.CO], 2019.
%H Leon Zaporski and Felix Flicker, <a href="https://arxiv.org/1811.00331">Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences</a>, arXiv:1811.00331 [nlin.CD], 2018.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-1).
%F G.f.: 1/(1 + 4*x + x^2).
%F a(n) = (-1)^n*A001353(n+1) = (-1)^(n + 1)*A106707(n+1).
%F From _Franck Maminirina Ramaharo_, Nov 11 2018: (Start)
%F a(n) = (-2)^n*((1 + sqrt(3)/2)^(n + 1) - (1 - sqrt(3)/2)^(n + 1))/sqrt(3).
%F E.g.f.: exp(-2*x)*(3*cosh(sqrt(3)*x) - 2*sqrt(3)*sinh(sqrt(3)*x))/3. (End)
%F a(n) = (-2)^n*Product_{k=1..n}(2 + cos(k*Pi/(n+1))). - _Peter Luschny_, Nov 28 2019
%F Sum_{k=0..n} a(k) = (1/6)*(1+a(n)-a(n+1)). - _Prabha Sivaramannair_, Sep 18 2023
%t CoefficientList[Series[1/(1+4*x+x^2),{x,0,50}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)
%o (Magma) I:=[1, -4]; [n le 2 select I[n] else -4*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 28 2012
%o (PARI) x='x+O('x^30); Vec(1/(1+4*x+x^2)) \\ _G. C. Greubel_, Feb 05 2018
%Y Cf. A001353, A106707.
%K easy,sign
%O 0,2
%A _Philippe Deléham_, Feb 04 2007
%E Typo in a(22) corrected by Neven Juric, Dec 20 2010
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