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

Author

Paul Curtz, Aug 27 2007

Keywords

• sign
• easy

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

Examples

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

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

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

Crossrefs

Cf. A257075, A291041.

Programs

Magma

&cat [[1, 1, -1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

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 *)

PARI

a(n)=1-2*(n%3==2) /* Jaume Oliver Lafont, Mar 24 2009 */

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

E.g.f.: (exp(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/3. - Stefano Spezia, Oct 19 2024

Extensions

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