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%I #26 Mar 12 2021 22:24:41
%S 1,0,0,2,0,2,2,0,0,0,2,0,2,2,0,0,0,0,0,2,2,2,0,0,2,0,2,4,0,0,0,0,0,0,
%T 0,2,0,0,2,0,2,0,2,0,0,6,0,0,2,0,0,0,2,0,4,0,0,0,0,2,0,2,0,0,0,0,0,0,
%U 0,4,2,0,0,0,0,6,2,0,0,0,2,0,0,2,2,0,0,0,0,0,6,2,0,0,0,0,2,0,0
%N Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 5 ].
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C The number of integer solutions (x, y) to n = 3*x^2 + 2*x*y + 5*y^2, discriminant -56. - _Ray Chandler_, Jul 12 2014
%H John Cannon, <a href="/A028928/b028928.txt">Table of n, a(n) for n = 0..10000</a>
%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%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 Expansion of phi(q^3) * phi(q^42) + 2*q^5 * chi(q) * psi(-q^3) * chi(q^14) * psi(-q^42) = phi(q^6) * phi(q^21) + 2*q^3 * chi(q^2) * psi(-q^6) * chi(q^7) * psi(-q^21) = phi(q^2) * phi(q^7) - 2*q^2 * phi(-q^4) * psi(q^7) * chi(-q) / chi(-q^28) in powers of q where phi(), psi(), chi() are Ramanujan theta functions - _Michael Somos_ and Alex Berkovich, Jun 06 2011
%F Expansion of - phi(q) * phi(q^14) + 2 * chi(q) * f(-q^7) * f(-q^8) * chi(q^14) in powers of q where phi(), chi(), f() are Ramanujan theta functions - _Michael Somos_, Jun 22 2011
%F G.f.: Sum_{n, m in Z} x^(3*n*n + 2*n*m + 5*m*m).
%e G.f. = 1 + 2*q^3 + 2*q^5 + 2*q^6 + 2*q^10 + 2*q^12 + 2*q^13 + 2*q^19 + 2*q^20 + 2*q^21 + 2*q^24 + 2*q^26 + 4*q^27 + 2*q^35 + 2*q^38 + 2*q^40 + 2*q^42 + 6*q^45 + ...
%t a[ n_] := If[ n < 1, Boole[ n == 0], If[ -1 != KroneckerSymbol[ -7, n / 7^IntegerExponent[ n, 7]], 0, Sum[ KroneckerSymbol[ -14, d], { d, Divisors @ n}]]]; (* _Michael Somos_, Jul 13 2011 *)
%o (PARI) {a(n) = if( n<1, n==0, qfrep([3, 1; 1, 5], n)[n] * 2)}; /* _Michael Somos_, Jun 06 2011 */
%o (PARI) {a(n) = if( n<1, n==0, (-1 == kronecker( -7, n / 7^valuation( n, 7))) * sumdiv( n, d, kronecker( -14, d)))}; /* _Michael Somos_, Jun 22 2011 */
%Y Cf. A106915.
%K nonn
%O 0,4
%A _N. J. A. Sloane_.
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