A112606 - OEIS

%I #27 Mar 12 2021 22:24:43

%S 1,1,0,1,0,0,3,2,0,2,1,0,2,0,0,1,2,0,0,0,0,3,0,0,2,2,0,4,1,0,2,0,0,0,

%T 4,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,0,2,2,0,2,3,0,2,0,0,4,2,0,0,2,0,1,0,

%U 0,4,0,0,2,0,0,2,0,0,1,2,0,0,2,0,2,0,0,0,0,0,4,1,0,0,0,0,2,4,0,4,0,0,4,0,0

%N Number of representations of n as a sum of six times a square and a triangular number.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C The greedy inverse starts 2, 0, 7, 6, 27, 300, 349, 14706, 216, 1035, 17107,... - _R. J. Mathar_, Apr 28 2020

%D M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

%H G. C. Greubel, <a href="/A112606/b112606.txt">Table of n, a(n) for n = 0..1000</a>

%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}(8n+1) - d_{2, 3}(8n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.

%F Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^12)^5 /(eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q. - _Michael Somos_, Sep 29 2006

%F Expansion of phi(q^6) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

%F Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -4, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -2, ...]. - _Michael Somos_, Sep 29 2006

%F G.f.: (Sum_{k} x^(6*k^2)) * (Sum_{k>0} x^((k^2-k)/2)). a(3*n+2)=0. - _Michael Somos_, Sep 29 2006

%F a(n) = A123484(24*n + 3) = A112604(2*n) = A112608(3*n). A131961(n) = a(3*n). A112608(n) = a(3*n + 1).

%e 1 + x + x^3 + 3*x^6 + 2*x^7 + 2*x^9 + x^10 + 2*x^12 + x^15 + 2*x^16 + ...

%e q + q^9 + q^25 + 3*q^49 + 2*q^57 + 2*q^73 + q^81 + 2*q^97 + q^121 + 2*q^129 + ...

%e a(6) = 3 since we can write 6 = 6*1^2 + 0 = 6*(-1)^2 + 0 = 0 + 6.

%t a[ n_] := If[ n < 0, 0, Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ 8 n + 1]}]] (* _Michael Somos_, Jun 16 2011 since V6 *)

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 3, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]] (* _Michael Somos_, Jun 16 2011 *)

%o (PARI) {a(n) = if( n<0, 0, n = 8*n + 1; sumdiv(n, d, kronecker(-3, 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)^2 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))} /* _Michael Somos_, Sep 29 2006 */

%Y Cf. A112604, A112608, A123484, A131961.

%K nonn

%O 0,7

%A _James A. Sellers_, Dec 21 2005