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%I #22 Feb 16 2023 21:33:40
%S 0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,
%T 1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,
%U 0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0,0,1,0,-1,0
%N Expansion of (x - x^3) / (1 - x^5) in powers of x.
%C Partial sums of A080891.
%H M. E. Muldoon and A. A. Ungar, <a href="http://www.jstor.org/stable/2691389">Beyond Sin and Cos</a>, Mathematics Magazine, 69,1,(1996).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1,-1).
%F E.g.f. : F(1, 5, 1, x)-F(1, 5, 3, x);
%F a(n) = Sum{k=0..n} Jacobi(k, 5).
%F Euler transform of length 5 sequence [ 0, -1, 0, 0, 1]. - _Michael Somos_, Mar 26 2012
%F G.f.: (x + x^2) / (1 + x + x^2 + x^3 + x^4).
%F a(n) = a(n + 5). a(-1 - n) = -a(n). a(5*n) = a(5*n + 2) = a(5*n+4) = 0. a(5*n + 1) = 1. a(5*n + 3) = -1. - _Michael Somos_, Mar 26 2012
%F a(n) = A010891(n-1)+A010891(n-2). - _R. J. Mathar_, Aug 11 2021
%e x - x^3 + x^6 - x^8 + x^11 - x^13 + x^16 - x^18 + x^21 - x^23 + x^26 + ...
%t LinearRecurrence[{-1,-1,-1,-1},{0,1,0,-1},120] (* _Harvey P. Dale_, Jan 14 2022 *)
%o (PARI) {a(n) = (n%5 == 1) - (n%5 == 3)} /* _Michael Somos_, Mar 26 2012 */
%Y Cf. A105385 (essentially the same).
%Y Cf. A080891 (first differences), A010891.
%K easy,sign
%O 0,1
%A _Paul Barry_, Feb 24 2004
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