A131713 Period 3: repeat [1, -2, 1].

1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1

Offset

0,2

Comments

Second differences of A131534. Binomial transform of 1, -3, 6, -9, 9, 0, ..., A057083 signed.

Nonsimple continued fraction expansion of sqrt(2)-1 = 0.414213562... - R. J. Mathar, Mar 08 2012

Links

Table of n, a(n) for n=0..77.

Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Formula

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^n) = 0^n, b(3^n) = 3 * 0^n - 2, b(p^n) = 1 if p > 3. - Michael Somos, Jan 02 2011

G.f.: (1-x)/(x^2+x+1). - R. J. Mathar, Nov 14 2007

a(n) = 2*cos((2n+1)*Pi/3). - Jaume Oliver Lafont, Nov 23 2008

a(n) = A117188(2*n). - R. J. Mathar, Jun 13 2011

a(n) + a(n-1) + a(n-2) = 0 for n>1, a(n) = a(n-3) for n>2. - Wesley Ivan Hurt, Jul 01 2016

a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*(-3)^k. - Peter Bala, Feb 06 2019

Maple

seq(op([1, -2, 1]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

Mathematica

f[n_] := If[ Mod[n, 3] == 1, -2, 1]; Array[f, 105, 0]
CoefficientList[Series[(1 - x)/(1 + x + x^2), {x, 0, 104}], x]
PadRight[{}, 120, {1, -2, 1}] (* Harvey P. Dale, Jan 25 2014 *)

Prog

(PARI) a(n)=[1, -2, 1][1+n%3] \\ Jaume Oliver Lafont, Mar 24 2009
(PARI) a(n)=1-3*(n%3==1) \\ Jaume Oliver Lafont, Mar 24 2009
(Magma) &cat [[1, -2, 1]^^30]; // Wesley Ivan Hurt, Jul 01 2016

Crossrefs

Cf. A057083, A061347, A131534.

Sequence in context: A100051 A281727 A122876 * A100063 A057079 A132419

Adjacent sequences: A131710 A131711 A131712 * A131714 A131715 A131716

Keyword

sign,easy,less

Author

Paul Curtz, Sep 14 2007

Extensions

Corrected and extended by Michael Somos, Jan 02 2011

Status

approved