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

1, -4, 15, -56, 209, -780, 2911, -10864, 40545, -151316, 564719, -2107560, 7865521, -29354524, 109552575, -408855776, 1525870529, -5694626340, 21252634831, -79315912984, 296011017105, -1104728155436, 4122901604639, -15386878263120, 57424611447841

Offset

0,2

Comments

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

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

References

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967, pp. 35-46.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

[Benoît] Clapeyron, Calcul d'une poutre élastique reposant librement sur des appuis inégalement espacés, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 45 (1857), 1076-1080.

Felix Flicker, Time quasilattices in dissipative dynamical systems, arXiv:1707.09371 [nlin.CD], 2017. Also SciPost Phys. 5, 001 (2018).

Pavel Galashin, Alexander Postnikov, and Lauren Williams, Higher secondary polytopes and regular plabic graphs, arXiv:1909.05435 [math.CO], 2019.

Leon Zaporski and Felix Flicker, Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences, arXiv:1811.00331 [nlin.CD], 2018.

Index entries for linear recurrences with constant coefficients, signature (-4,-1).

Formula

G.f.: 1/(1 + 4*x + x^2).

a(n) = (-1)^n*A001353(n+1) = (-1)^(n + 1)*A106707(n+1).

From Franck Maminirina Ramaharo, Nov 11 2018: (Start)

a(n) = (-2)^n*((1 + sqrt(3)/2)^(n + 1) - (1 - sqrt(3)/2)^(n + 1))/sqrt(3).

E.g.f.: exp(-2*x)*(3*cosh(sqrt(3)*x) - 2*sqrt(3)*sinh(sqrt(3)*x))/3. (End)

a(n) = (-2)^n*Product_{k=1..n}(2 + cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019

Sum_{k=0..n} a(k) = (1/6)*(1+a(n)-a(n+1)). - Prabha Sivaramannair, Sep 18 2023

Mathematica

CoefficientList[Series[1/(1+4*x+x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 28 2012 *)

Prog

(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
(PARI) x='x+O('x^30); Vec(1/(1+4*x+x^2)) \\ G. C. Greubel, Feb 05 2018

Crossrefs

Cf. A001353, A106707.

Sequence in context: A242495 A221859 A106707 * A195503 A001353 A010905

Adjacent sequences: A125902 A125903 A125904 * A125906 A125907 A125908

Keyword

easy,sign

Author

Philippe Deléham, Feb 04 2007

Extensions

Typo in a(22) corrected by Neven Juric, Dec 20 2010

Status

approved