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A125905 a(0) = 1, a(1) = -4, a(n) = -4*a(n-1) - a(n-2) for n > 1.

%I A125905 #71 Sep 19 2023 15:33:04
%S A125905 1,-4,15,-56,209,-780,2911,-10864,40545,-151316,564719,-2107560,
%T A125905 7865521,-29354524,109552575,-408855776,1525870529,-5694626340,
%U A125905 21252634831,-79315912984,296011017105,-1104728155436,4122901604639,-15386878263120,57424611447841
%N A125905 a(0) = 1, a(1) = -4, a(n) = -4*a(n-1) - a(n-2) for n > 1.
%C A125905 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 A125905 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 A125905 Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967, pp. 35-46.
%H A125905 Vincenzo Librandi, <a href="/A125905/b125905.txt">Table of n, a(n) for n = 0..1000</a>
%H A125905 [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 A125905 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 A125905 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 A125905 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 A125905 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-1).
%F A125905 G.f.: 1/(1 + 4*x + x^2).
%F A125905 a(n) = (-1)^n*A001353(n+1) = (-1)^(n + 1)*A106707(n+1).
%F A125905 From _Franck Maminirina Ramaharo_, Nov 11 2018: (Start)
%F A125905 a(n) = (-2)^n*((1 + sqrt(3)/2)^(n + 1) - (1 - sqrt(3)/2)^(n + 1))/sqrt(3).
%F A125905 E.g.f.: exp(-2*x)*(3*cosh(sqrt(3)*x) - 2*sqrt(3)*sinh(sqrt(3)*x))/3. (End)
%F A125905 a(n) = (-2)^n*Product_{k=1..n}(2 + cos(k*Pi/(n+1))). - _Peter Luschny_, Nov 28 2019
%F A125905 Sum_{k=0..n} a(k) = (1/6)*(1+a(n)-a(n+1)). - _Prabha Sivaramannair_, Sep 18 2023
%t A125905 CoefficientList[Series[1/(1+4*x+x^2),{x,0,50}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)
%o A125905 (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 A125905 (PARI) x='x+O('x^30); Vec(1/(1+4*x+x^2)) \\ _G. C. Greubel_, Feb 05 2018
%Y A125905 Cf. A001353, A106707.
%K A125905 easy,sign
%O A125905 0,2
%A A125905 _Philippe Deléham_, Feb 04 2007
%E A125905 Typo in a(22) corrected by Neven Juric, Dec 20 2010