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A014458
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Theta series of quadratic form with Gram matrix [ 2, 1, 0; 1, 4, 1; 0, 1, 2 ].
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2
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1, 4, 12, 0, 20, 8, 10, 8, 28, 4, 32, 16, 0, 8, 40, 8, 52, 8, 12, 16, 40, 0, 40, 16, 26, 20, 64, 0, 40, 24, 0, 24, 60, 8, 72, 16, 20, 24, 40, 0, 64, 24, 32, 16, 80, 8, 80, 16, 0, 28, 84, 16, 40, 24, 10, 16, 104, 0, 96, 32, 40, 24, 40, 8, 116, 32, 0, 32, 40, 16, 80, 48, 28, 16, 128, 0, 80
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OFFSET
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0,2
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COMMENTS
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This is the tetragonal I lattice (the even holotype) of dimension 3.
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LINKS
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FORMULA
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a(n) = coefficient of q^2n in theta3(q)^2*theta3(q^3). Theta series of even sublattice of Z^2+sqrt(3)Z - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 07 2002
Expansion of (phi(q)^2 * phi(q^3) + phi(-q)^2 * phi(-q^3)) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Apr 05 2012
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EXAMPLE
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1 + 4*x + 12*x^2 + 20*x^4 + 8*x^5 + 10*x^6 + 8*x^7 + 28*x^8 + 4*x^9 + ...
1 + 4*q^2 + 12*q^4 + 20*q^8 + 8*q^10 + 10*q^12 + 8*q^14 + 28*q^16 + 4*q^18 + ...
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MATHEMATICA
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terms = 77; s = Normal[EllipticTheta[3, 0, q]^2*EllipticTheta[3, 0, q^3] + O[q]^(3*terms)][[1 ;; 2 terms]]; Partition[CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 04 2017 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, qfrep( [ 2, 1, 0; 1, 4, 1; 0, 1, 2], n, 1)[n] * 2 )} /* Michael Somos, Apr 05 2012 */
(PARI) {a(n) = if( n<1, n==0, qfrep( [ 1, 0, 0; 0, 1, 0; 0, 0, 3], n, 1)[n] * 2 )} /* Michael Somos, Apr 05 2012 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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