A122572 a(n) = -14*a(n-1) - a(n-2), with a(1) = a(2) = 1.

1, 1, -15, 209, -2911, 40545, -564719, 7865521, -109552575, 1525870529, -21252634831, 296011017105, -4122901604639, 57424611447841, -799821658665135, 11140078609864049, -155161278879431551, 2161117825702177665, -30100488280951055759, 419245718107612602961

Offset: 1

Author

Roger L. Bagula, Sep 17 2006

Keywords

• sign
• easy

Comments

Characteristic polynomial associated with the elliptic cubic invariant x^8 + 14*x^4 + 1.

References

• Henry MacKean and Victor Moll, Elliptic Curves, Cambridge University Press, New York, 1997, page 22

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..874
• Gareth Jones and David Singerman, Belyi Functions, Hypermaps and Galois Groups, Bull. London Math. Soc., 28 (1996), 561-590.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (-14,-1).

Crossrefs

Cf. A028230, A122571.

Programs

Magma

[n le 2 select 1 else -14*Self(n-1) -Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 29 2024

Mathematica

LinearRecurrence[{-14,-1},{1,1},30] (* Harvey P. Dale, Jul 30 2013 *)

SageMath

A122572=BinaryRecurrenceSequence(-14,-1,1,1)
[A122572(n-1) for n in range(1,41)] # G. C. Greubel, Oct 29 2024

Formula

G.f.: x*(1+15*x)/(1+14*x+x^2). - Philippe Deléham, Nov 16 2008 [Corrected by Richard Choulet, Nov 21 2008]

a(n) = ((3+2*sqrt(3))/6)*(-7+4*sqrt(3))^(n-1)+((3-2*sqrt(3))/6)*(-7-4*sqrt(3))^(n-1) (n>=1). - Richard Choulet, Nov 21 2008

a(n) = (-1)^n*A028230(n-1), n>1. - R. J. Mathar, Mar 19 2009

a(n) = b such that (-1)^(2*n-3)*Integral_{x=0..Pi/2} cos((2*n-3)*x)/(2+sin(x)) dx = c + b*(log(2)-log(3)). - Francesco Daddi, Aug 01 2011

E.g.f.: 15 - exp(-7*x)*( 15*cosh(4*sqrt(3)*x) + (26*sqrt(3)/3)*sinh(4*sqrt(3)*x) ). - G. C. Greubel, Oct 29 2024

Extensions

Edited by N. J. A. Sloane, Dec 04 2006