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%I #58 Dec 18 2023 12:18:35
%S 1,-1,3,-7,17,-41,99,-239,577,-1393,3363,-8119,19601,-47321,114243,
%T -275807,665857,-1607521,3880899,-9369319,22619537,-54608393,
%U 131836323,-318281039,768398401,-1855077841,4478554083,-10812186007,26102926097,-63018038201,152139002499
%N a(n) = -2*a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=-1.
%C Inverse binomial transform of A077957.
%C 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
%C 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
%C 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
%C 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
%C 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
%H Harvey P. Dale, <a href="/A123335/b123335.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,1).
%F a(n) = (-1)^n*A001333(n).
%F G.f.: (1+x)/(1+2*x-x^2).
%F a(n) = A077985(n) + A077985(n-1). - _R. J. Mathar_, Mar 28 2011
%F 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
%F G.f.: 1/(1 + x/(1 + 2*x/(1 - x))). - _Michael Somos_, Apr 19 2022
%F E.g.f.: exp(-x)*cosh(sqrt(2)*x). - _Stefano Spezia_, Feb 01 2023
%e G.f. = 1 - x + 3*x^2 - 7*x^3 + 17*x^4 - 41*x^5 + 99*x^6 + ... - _Michael Somos_, Apr 19 2022
%p a:= n-> (M-> M[2, 1]+M[2, 2])(<<2|1>, <1|0>>^(-n)):
%p seq(a(n), n=0..33); # _Alois P. Heinz_, Jun 22 2021
%t LinearRecurrence[{-2,1},{1,-1},40] (* _Harvey P. Dale_, Nov 03 2011 *)
%o (PARI) x='x+O('x^50); Vec((1+x)/(1+2*x-x^2)) \\ _G. C. Greubel_, Oct 12 2017
%o (PARI) {a(n) = real((-1 + quadgen(8))^n)}; /* _Michael Somos_, Apr 19 2022 */
%o (Magma) [Round(1/2*((-1-Sqrt(2))^n+(-1+Sqrt(2))^n)): n in [0..30]]; // _G. C. Greubel_, Oct 12 2017
%Y Cf. A000129, A001333, A077985, A251732, A001541 (bisection), A002315 (bisection).
%K sign,easy
%O 0,3
%A _Philippe Deléham_, Jun 27 2007
%E Corrected by _N. J. A. Sloane_, Oct 05 2008
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