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

Author

Philippe Deléham, Feb 04 2007

Keywords

• easy
• sign

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).

Crossrefs

Cf. A001353, A106707.

Programs

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

Mathematica

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

PARI

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

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

Extensions

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