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A028928
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Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 5 ].
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3
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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, 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, 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
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OFFSET
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0,4
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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
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LINKS
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John Cannon, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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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
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
G.f.: Sum_{n, m in Z} x^(3*n*n + 2*n*m + 5*m*m).
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, qfrep([3, 1; 1, 5], n)[n] * 2)}; /* Michael Somos, Jun 06 2011 */
(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 */
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CROSSREFS
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Cf. A106915.
Sequence in context: A298141 A160210 A174610 * A260149 A091379 A151758
Adjacent sequences: A028925 A028926 A028927 * A028929 A028930 A028931
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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