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
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
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
KEYWORD
sign,easy
AUTHOR
Paul Curtz, Aug 27 2007
EXTENSIONS
Edited by N. J. A. Sloane, Sep 15 2007
STATUS
approved