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A008653
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Theta series of direct sum of 2 copies of hexagonal lattice.
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3
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1, 12, 36, 12, 84, 72, 36, 96, 180, 12, 216, 144, 84, 168, 288, 72, 372, 216, 36, 240, 504, 96, 432, 288, 180, 372, 504, 12, 672, 360, 216, 384, 756, 144, 648, 576, 84, 456, 720, 168, 1080, 504, 288, 528, 1008, 72, 864, 576, 372, 684, 1116, 216, 1176, 648, 36
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 460, Entry 3(i).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.
Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.
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LINKS
| Michael Gilleland, Some Self-Similar Integer Sequences
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
| Expansion of (theta_3(z)*theta_3(3z)+theta_2(z)*theta_2(3z))^2.
Expansion of a(q)^2 in powers of q where a() is a cubic AGM analog function.
G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w)= u^2 +9*v^2 +16*w^2 -6*u*v +4*u*w -24*v*w . - Michael Somos, Jul 19 2004
G.f.: 1 +12* Sum_{k>0} x^k/ (1-x^k)^2 -36* Sum_{k>0} x^(3k)/ (1-x^(3k))^2 . - Michael Somos Apr 15 2007
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PROG
| (PARI) a(n)=if(n<1, n==0, 12*(sigma(3*n)-3*sigma(n))) /* Michael Somos, Jul 19 2004 */
(PARI) a(n)=if(n<0, 0, polcoeff(sum(k=1, n, 6*x^k/(1+x^k+x^(2*k)), 1+x^n*O(x))^2, n)) /* Michael Somos, Jul 19 2004 */
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CROSSREFS
| a(n)=12*A046913(n) unless n=0. Convolution square of A004016.
Sequence in context: A009649 A195539 A007794 * A038006 A205967 A203378
Adjacent sequences: A008650 A008651 A008652 * A008654 A008655 A008656
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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