OFFSET
0,3
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
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,1).
FORMULA
a(n) = (-1)^n*A001333(n).
G.f.: (1+x)/(1+2*x-x^2).
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
EXAMPLE
G.f. = 1 - x + 3*x^2 - 7*x^3 + 17*x^4 - 41*x^5 + 99*x^6 + ... - Michael Somos, Apr 19 2022
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 *)
PROG
(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 */
(Magma) [Round(1/2*((-1-Sqrt(2))^n+(-1+Sqrt(2))^n)): n in [0..30]]; // G. C. Greubel, Oct 12 2017
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Philippe Deléham, Jun 27 2007
EXTENSIONS
Corrected by N. J. A. Sloane, Oct 05 2008
STATUS
approved