Search: id:A008653
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%I A008653
%S A008653 1,12,36,12,84,72,36,96,180,12,216,144,84,168,288,72,372,216,36,240,504,
%T A008653 96,432,288,180,372,504,12,672,360,216,384,756,144,648,576,84,456,720,
%U A008653 168,1080,504,288,528,1008,72,864,576,372,684,1116,216,1176,648,36
%N A008653 Theta series of direct sum of 2 copies of hexagonal lattice.
%C A008653 The hexagonal lattice is the familiar 2-dimensional lattice in which 
               each point has 6 neighbors. This is sometimes called the triangular 
               lattice.
%D A008653 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 
               460, Entry 3(i).
%D A008653 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 110.
%H A008653 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%H A008653 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/A2.html">Home page for hexagonal (or triangular) lattice 
               A2</a>
%F A008653 Expansion of (theta_3(z)*theta_3(3z)+theta_2(z)*theta_2(3z))^2.
%F A008653 Expansion of a(q)^2 in powers of q where a() is a cubic AGM analog function.
%F A008653 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
%F A008653 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
%o A008653 (PARI) a(n)=if(n<1,n==0,12*(sigma(3*n)-3*sigma(n))) /* Michael Somos, 
               Jul 19 2004 */
%o A008653 (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 */
%Y A008653 a(n)=12*A046913(n) unless n=0. Convolution square of A004016.
%Y A008653 Sequence in context: A103472 A009649 A007794 this_sequence A038006 A073543 
               A074234
%Y A008653 Adjacent sequences: A008650 A008651 A008652 this_sequence A008654 A008655 
               A008656
%K A008653 nonn,easy
%O A008653 0,2
%A A008653 N. J. A. Sloane (njas(AT)research.att.com).

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