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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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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
| John Cannon, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| 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} x ^ (3*n*n + 2*n*m + 5*m*m).
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EXAMPLE
| 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
| 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
| (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
| Sequence in context: A196517 A160210 A174610 * A091379 A151758 A164272
Adjacent sequences: A028925 A028926 A028927 * A028929 A028930 A028931
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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