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%I #26 Mar 12 2021 22:24:43
%S 1,3,2,1,4,2,1,4,0,2,5,2,2,0,2,3,4,2,0,6,0,1,4,0,2,4,4,0,3,2,2,4,2,0,
%T 0,2,3,8,0,2,4,0,2,0,2,3,6,0,0,4,2,2,4,2,2,3,2,2,0,4,0,4,0,0,8,2,1,4,
%U 0,0,8,2,2,0,2,2,0,2,1,4,2,4,6,0,2,4,0,4,0,0,0,7,4,0,4,2,2,0,0,0,6,2,4,4,2
%N Number of representations of n as the sum of a square and a triangular number.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A112603/b112603.txt">Table of n, a(n) for n = 0..5000</a>
%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F a(n) = A002325(8n+1). [Hirschhorn]
%F Expansion of q^(-1/8) * eta(q^2)^7 / (eta(q)^3 * eta(q^4)^2) in powers of q. - _Michael Somos_, Sep 29 2006
%F Expansion of phi(q) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Sep 29 2006
%F Euler transform of period 4 sequence [ 3, -4, 3, -2, ...]. - _Michael Somos_, Sep 29 2006
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A139093. - _Michael Somos_, Mar 16 2011
%F G.f.: (Sum_{k} x^(k^2)) * (Sum_{k>0} x^((k^2 - k)/2)). - _Michael Somos_, Sep 29 2006
%e a(4) = 4 since we can write 4 = 2^2 + 0 = (-2)^2 + 0 = 1^2 + 3 = (-1)^2 + 3.
%e 1 + 3*x + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + x^6 + 4*x^7 + 2*x^9 + 5*x^10 + ...
%e q + 3*q^9 + 2*q^17 + q^25 + 4*q^33 + 2*q^41 + q^49 + 4*q^57 + 2*q^73 + ...
%t a[n_] := DivisorSum[8n + 1, KroneckerSymbol[-2, #]&]; Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Dec 06 2015, adapted from PARI *)
%o (PARI) {a(n) = if( n<0, 0, n = 8*n + 1; sumdiv( n, d, kronecker( -2, d)))} /* _Michael Somos_, Sep 29 2006 */
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 /(eta(x + A)^3 * eta(x^4 + A)^2), n))} /* _Michael Somos_, Sep 29 2006 */
%Y Cf. A139093.
%K nonn
%O 0,2
%A _James A. Sellers_, Dec 21 2005
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