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%I #24 Dec 28 2022 14:37:12
%S 1,2,0,0,2,0,1,2,0,2,2,0,0,0,0,2,2,0,1,2,0,0,4,0,0,2,0,2,0,0,0,2,0,0,
%T 2,0,3,2,0,0,2,0,2,2,0,2,0,0,0,2,0,0,2,0,2,2,0,0,0,0,1,4,0,0,4,0,0,2,
%U 0,2,2,0,2,0,0,0,2,0,0,0,0,2,2,0,0,4,0,2,0,0,1,2,0,0,2,0,2,0,0,4,4,0,0,0,0
%N Number of representations of n as a sum of a square and six times a triangular number.
%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H G. C. Greubel, <a href="/A112605/b112605.txt">Table of n, a(n) for n = 0..1000</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) = d_{1, 3}(4n+3) - d_{2, 3}(4n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
%F Expansion of q^(-3/4)*eta(q^2)^5*eta(q^12)^2/(eta(q)^2*eta(q^4)^2*eta(q^6)) in powers of q. - _Michael Somos_, May 20 2006
%F Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -2, 2, -1, 2, -3, 2, -2, ...]. - _Michael Somos_, May 20 2006
%F a(n)=A002324(4n+3). - _Michael Somos_, May 20 2006
%F Expansion of phi(q)*psi(q^6) in powers of q where phi(),psi() are Ramanujan theta functions. - _Michael Somos_, May 20 2006, Sep 29 2006
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A164273. - _Michael Somos_, Aug 11 2009
%F a(3*n + 2) = 0. - _Michael Somos_, Aug 11 2009
%e a(22) = 4 since we can write 22 = 4^2 + 6*1 = (-4)^2 + 6*1 = 2^2 + 6*3 = (-2)^2 + 6*3.
%e G.f. = 1 + 2*x + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^15 + 2*x^16 + ... - _Michael Somos_, Aug 11 2009
%e G.f. = q^3 + 2*q^7 + 2*q^19 + q^27 + 2*q^31 + 2*q^39 + 2*q^43 + 2*q^63 + ... - _Michael Somos_, Aug 11 2009
%t a[n_] := DivisorSum[4n+3, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Dec 04 2015, adapted from PARI *)
%o (PARI) {a(n) = if(n<0, 0, sumdiv(4*n+3, d, kronecker(-3, d)))}; /* _Michael Somos_, May 20 2006 */
%o (PARI) {a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^12+A)^2 / eta(x+A)^2 / eta(x^4+A)^2 / eta(x^6+A), n))}; /* _Michael Somos_, May 20 2006 */
%Y A112608(n) = a(2*n). 2 * A112609(n) = a(2*n + 1). A112604(n) = a(3*n). 2 * A121361(n) = a(3*n + 1). A112606(n) = a(6*n). 2 * A131962(n) = a(6*n + 1). 2 * A112607(n) = a(6*n + 3). 2 * A131964(n) = a(6*n + 4). - _Michael Somos_, Aug 11 2009
%K nonn
%O 0,2
%A _James A. Sellers_, Dec 21 2005
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