A123335 a(n) = -2*a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=-1.

1, -1, 3, -7, 17, -41, 99, -239, 577, -1393, 3363, -8119, 19601, -47321, 114243, -275807, 665857, -1607521, 3880899, -9369319, 22619537, -54608393, 131836323, -318281039, 768398401, -1855077841, 4478554083, -10812186007, 26102926097, -63018038201, 152139002499

Offset: 0

Author

Philippe Deléham, Jun 27 2007

Keywords

• sign
• easy

Comments

Inverse binomial transform of A077957.

The inverse of the g.f. is 3-x-2/(1+x) which generates 1, 1, -2, +2, -2, +2, ... (-2, +2 periodically continued). - Gary W. Adamson, Jan 10 2011

Pisano period lengths: 1, 1, 8, 4, 12, 8, 6, 4, 24, 12, 24, 8, 28, 6, 24, 8, 16, 24, 40, 12, ... - R. J. Mathar, Aug 10 2012

a(n) is the rational part of the Q(sqrt(2)) integer (sqrt(2) - 1)^n = a(n) + A077985(n-1)*sqrt(2), with A077985(-1) = 0. - Wolfdieter Lang, Dec 07 2014

3^n*a(n) = A251732(n) gives the rational part of the integer in Q(sqrt(2)) giving the length of a variant of Lévy's C-curve at iteration step n. - Wolfdieter Lang, Dec 07 2014

Define u(0) = 1/0, u(1) = -1/1, and u(n) = -(8 + 3*u(n-1)*u(n-2))/(3*u(n-1) + 2*u(n-2)) for n>1. Then u(n) = a(n)/A000219(n). - Michael Somos, Apr 19 2022

Examples

			G.f. = 1 - x + 3*x^2 - 7*x^3 + 17*x^4 - 41*x^5 + 99*x^6 + ... - _Michael Somos_, Apr 19 2022
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,1).

Crossrefs

Cf. A000129, A001333, A077985, A251732, A001541 (bisection), A002315 (bisection).

Programs

Magma

[Round(1/2*((-1-Sqrt(2))^n+(-1+Sqrt(2))^n)): n in [0..30]]; // G. C. Greubel, Oct 12 2017

Maple

a:= n-> (M-> M[2, 1]+M[2, 2])(<<2|1>, <1|0>>^(-n)):
seq(a(n), n=0..33);  # Alois P. Heinz, Jun 22 2021

Mathematica

LinearRecurrence[{-2,1},{1,-1},40] (* Harvey P. Dale, Nov 03 2011 *)

PARI

x='x+O('x^50); Vec((1+x)/(1+2*x-x^2)) \\ G. C. Greubel, Oct 12 2017

PARI

{a(n) = real((-1 + quadgen(8))^n)}; /* Michael Somos, Apr 19 2022 */

Formula

a(n) = (-1)^n*A001333(n).

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

a(n) = A077985(n) + A077985(n-1). - R. J. Mathar, Mar 28 2011

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 19 2013

G.f.: 1/(1 + x/(1 + 2*x/(1 - x))). - Michael Somos, Apr 19 2022

E.g.f.: exp(-x)*cosh(sqrt(2)*x). - Stefano Spezia, Feb 01 2023

Extensions

Corrected by N. J. A. Sloane, Oct 05 2008