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A123063
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Theta series of lattice with Gram matrix [4,1;1,8].
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4
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1, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 2, 2, 4, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 4, 0, 0, 2, 2, 0, 2, 0, 6, 2, 0, 0, 0, 2, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 6, 0, 0, 2, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0, 6, 2, 2, 0, 0, 0, 4, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 2, 4, 0, 6, 2, 0, 2, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = number of solutions to n = 2*x^2 + x*y + 4*y^2 in integers, hence a(n) nonzero if and only if n is in A123064 and p is prime and a(p) = 2 if and only if p is in A106872. - Michael Somos, Jul 16 2011
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REFERENCES
| J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 82.
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FORMULA
| G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = +u6*u1^3 + 2*u3*u2^3 - 3*u3^3*u2 - 6*u6^3*u1 + 6*u6*u2^2*u1 - 6*u3*u2^2*u1 + 3*u3*u2*u1^2 - 6*u6*u2*u1^2 - 9*u6*u3^2*u1 - 18*u6^2*u3*u2 + 18*u6*u3^2*u2 + 18*u6^2*u3*u1. - Michael Somos Sep 28 2006
G.f. is a period 1 Fourier series which satisfies f(-1/(31*t)) = 31^(1/2)*(t/i)*f(t) where q = exp(2*pi*i*t). - Michael Somos, Jul 16 2011
G.f.: Sum_{n,m} x^(2*n^2 + n*m + 4*m^2).
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EXAMPLE
| 1 + 2*x^2 + 2*x^4 + 2*x^5 + 2*x^7 + 2*x^8 + 2*x^10 + 2*x^14 + 4*x^16 + 2*x^18 + ...
1 + 2*q^4 + 2*q^8 + 2*q^10 + 2*q^14 + 2*q^16 + 2*q^20 + 2*q^28 + 4*q^32 + 2*q^36 + ...
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PROG
| (MAGMA) L:=LatticeWithGram(2, [4, 1, 1, 8] ); T<q> := ThetaSeries(L, 500); T;
(PARI) {a(n) = if( n<1, n==0, qfrep( [4, 1; 1, 8], n, 1)[n] * 2)} /* Michael Somos, Sep 28 2006 */
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CROSSREFS
| Cf. A106872, A123064, A123065.
Sequence in context: A024713 A123530 A161516 * A031358 A029317 A127800
Adjacent sequences: A123060 A123061 A123062 * A123064 A123065 A123066
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 27 2006
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