Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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