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Search: id:A014455
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| A014455 |
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Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 1, 0; 0, 0, 2 ]. |
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+0
2
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| 1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 24, 24, 0, 16, 12, 16, 30, 24, 24, 16, 24, 16, 8, 28, 24, 32, 48, 8, 0, 32, 6, 32, 48, 16, 36, 40, 24, 16, 24, 16, 48, 40, 24, 40, 0, 32, 24, 36, 30, 16, 72, 24, 32, 48, 0, 32, 72, 24, 48, 40, 0, 48, 12, 16, 48, 56, 48, 32, 48, 16, 30, 64
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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This is the tetragonal P lattice (the classical holotype) of dimension 3.
Number of integer solutions to x^2+y^2+2z^2=n.
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LINKS
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John Cannon, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
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FORMULA
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Expansion of eta(q^2)^8*eta(q^4)/(eta(q)^4*eta(q^8)^2) in powers of q. - Michael Somos Jul 05 2005
Euler transform of period 8 sequence [4, -4, 4, -5, 4, -4, 4, -3, ...]. - Michael Somos, Jul 07 2005
G.f.: theta_3(q)^2*theta_3(q^2) = Product_{k>0} (1-x^(2k))^8*(1-x^(4k))/((1-x^k)^4*(1-x^(8k))^2).
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EXAMPLE
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1 + 4*q + 6*q^2 + 8*q^3 + 12*q^4 + 8*q^5 + 8*q^6 + 16*q^7 + 6*q^8 + 12*q^9 + ...
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PROGRAM
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(PARI) a(n)=if(n<1, n==0, 2*qfrep([1, 0, 0; 0, 1, 0; 0, 0, 2], n)[n]) /* Michael Somos Jul 05 2005 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^8*eta(x^4+A)/(eta(x+A)^4*eta(x^8+A)^2), n))} /* Michael Somos Jul 05 2005 */
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CROSSREFS
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Sequence in context: A075325 A026278 A139404 this_sequence A110646 A031359 A110606
Adjacent sequences: A014452 A014453 A014454 this_sequence A014456 A014457 A014458
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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