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A004020
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Theta series of square lattice with respect to edge.
(Formerly M0931)
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7
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2, 4, 2, 4, 4, 0, 6, 4, 0, 4, 4, 4, 2, 4, 0, 4, 8, 0, 4, 0, 2, 8, 4, 0, 4, 4, 0, 4, 4, 4, 2, 8, 0, 0, 4, 0, 8, 4, 4, 4, 0, 0, 6, 4, 0, 4, 8, 0, 4, 4, 0, 8, 0, 0, 0, 8, 6, 4, 4, 0, 4, 4, 0, 0, 4, 4, 8, 4, 0, 4, 4, 0, 6, 4, 0, 0, 8, 0, 4, 4, 0, 12, 0, 4, 4, 0, 0, 4, 4, 0, 2, 8, 4, 4, 8, 0, 0, 4, 0, 4, 4, 4, 4, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
Number of solutions in integers of n = x^2+y^2+y.
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: 2(Sum_{k>=0} x^((k^2+k)/2))^2 = (Sum_k x^(k^2+k))(Sum_k x^(k^2)).
Expansion of q^(-1/2)c(q)/2 in powers of q^2 where c(q) is the third function in the quadratic Gauss AGM. - Michael Somos, Feb 10 2006
Expansion of 2 * phi(q) * psi(q^2) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 10 2006
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PROG
| (PARI) a(n)=local(X); if(n<0, 0, X=x+x*O(x^n); 2*polcoeff(eta(X^2)^4/eta(X)^2, n))
(PARI) a(n)=2*if(n<1, n==0, polcoeff(sum(k=0, (sqrtint(8*n+1)-1)\2, x^(k*(k+1)/2), x*O(x^n))^2, n))
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CROSSREFS
| a(n)=2*A008441(n)=A004531(4n+1).
Sequence in context: A032059 A074075 A184186 * A143235 A069465 A047947
Adjacent sequences: A004017 A004018 A004019 * A004021 A004022 A004023
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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