A131561 Period 3: repeat [1, 1, -1].

1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1

Offset

0,1

Comments

Other than the first term, this sequence represents numerators in a fraction expansion of log(2) - Pi/8. - Mohammad K. Azarian, Sep 27 2011

Also, the arithmetic function uhat(n,3,3) as defined in A291041. - Robert Price, Aug 25 2017

References

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

Links

Table of n, a(n) for n=0..109.

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Formula

a(n) = (4*cos((2*n - 1) * Pi/3) + 1) / 3. - Federico Acha Neckar (f0383864(AT)hotmail.com), Sep 02 2007

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

G.f.: (1+x-x^2)/(1-x^3). - Jaume Oliver Lafont, Mar 24 2009

a(n) = (-1)^((n-1) mod 3). - Christopher Richmond, Oct 07 2011

a(n) = a(n-1)^2 - a(n-1) - a(n-2), for a(0),a(1) = 1,1; or same repeating pattern with 1,-1 or -1,1 as initial values. - Richard R. Forberg, Jun 13 2013

a(n+1) = A257075(n) for all n in Z. - Michael Somos, May 13 2015

a(n) = a(n-3) for n>2. - Wesley Ivan Hurt, Jul 02 2016

Product_{n >= 1} (1 + a(n-1)*x^n) = 1 + x + x^2 + x^5 + x^7 + x^12 + x^15 + ... = Sum_{n >= 0} x^A001318(n), a companion identity to Euler's pentagonal number theorem. - Peter Bala, Aug 30 2017

Example

G.f. = 1 + x - x^2 + x^3 + x^4 - x^5 + x^6 + x^7 - x^8 + x^9 + x^10 + ...

Maple

A131561 := proc(n) op((n mod 3)+1, [1, 1, -1]) ; end: seq(A131561(n), n=0..120); # R. J. Mathar, Oct 18 2007

Mathematica

Table[(-1)^Mod[n-1, 3], {n, 0, 120}] (* Michael De Vlieger, Mar 07 2015 *)
PadRight[{}, 120, {1, 1, -1}] (* Harvey P. Dale, Mar 15 2021 *)

Prog

(PARI) a(n)=1-2*(n%3==2) /* Jaume Oliver Lafont, Mar 24 2009 */
(Magma) &cat [[1, 1, -1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

Crossrefs

Cf. A257075, A291041.

Sequence in context: A362928 A071935 A096809 * A110515 A355437 A071936

Adjacent sequences: A131558 A131559 A131560 * A131562 A131563 A131564

Keyword

sign,easy

Author

Paul Curtz, Aug 27 2007

Extensions

Edited by N. J. A. Sloane, Sep 15 2007

Status

approved