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A214505 a(n) = 1 if n is four times a triangular number, -1 if one more than twelve times a triangular number else 0.

%I A214505 #19 Aug 30 2025 17:11:22
%S A214505 1,-1,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
%T A214505 0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%U A214505 0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1
%N A214505 a(n) = 1 if n is four times a triangular number, -1 if one more than twelve times a triangular number else 0.
%C A214505 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A214505 Antti Karttunen, <a href="/A214505/b214505.txt">Table of n, a(n) for n = 0..65537</a>
%H A214505 S. Cooper and M. Hirschhorn, <a href="http://dx.doi.org/10.1216/rmjm/1008959672">On some infinite product identities</a>, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 5.
%H A214505 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214505 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214505 Expansion of psi(x^4) - x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.
%F A214505 Expansion of f(-x, x^5) * f(-x^4, -x^8) / f(x, -x) in powers of x where f(,) is the Ramanujan two-variable theta function.
%F A214505 Euler transform of period 24 sequence [ -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, ...].
%F A214505 G.f.: (Sum_{k} x^(2*k*(k + 1)) - x^(6*k*(k + 1) + 1)) / 2.
%F A214505 a(n) = A214295(2*n + 1).
%e A214505 G.f. = 1 - x + x^4 + x^12 - x^13 + x^24 - x^37 + x^40 + x^60 - x^73 + x^84 + ...
%e A214505 G.f. = q - q^3 + q^9 + q^25 - q^27 + q^49 - q^75 + q^81 + q^121 - q^147 + q^169 + ...
%o A214505 (PARI) {a(n) = n = 2*n + 1; issquare(n) - issquare(3*n)};
%Y A214505 Cf. A010052, A214295.
%K A214505 sign,changed
%O A214505 0,1
%A A214505 _Michael Somos_, Jul 19 2012