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A097057
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Number of integer solutions to a^2 + b^2 + 2*c^2 + 2*d^2 = n.
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3
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1, 4, 8, 16, 24, 24, 32, 32, 24, 52, 48, 48, 96, 56, 64, 96, 24, 72, 104, 80, 144, 128, 96, 96, 96, 124, 112, 160, 192, 120, 192, 128, 24, 192, 144, 192, 312, 152, 160, 224, 144, 168, 256, 176, 288, 312, 192, 192, 96, 228, 248, 288, 336, 216, 320, 288, 192, 320, 240, 240
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| One of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos Apr 01 2008
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 31.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.29).
S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917) 11-21)
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LINKS
| Y. Mimura, Almost Universal Quadratic Forms.
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FORMULA
| Euler transform of period 8 sequence [4, -2, 4, -8, 4, -2, 4, -4, ...]. - Michael Somos, Sep 17 2004
Multiplicative with a(n) = 4*b(n), b(2) = 2, b(2^e) = 6 if e>1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>2. - Michael Somos, Sep 17 2004
Expansion of (eta(q^2) * eta(q^4))^6 / (eta(q) * eta(q^8))^4 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8 (t/i)^2 f(t) where q = exp(2 pi i t). - Michael Somos Jul 05 2011
G.f.: (theta_3(q) * theta_3(q^2))^2.
G.f.: Product_{k>0} ((1-x^(2k))(1-x^(4k)))^6/((1-x^k)(1-x^(8k)))^4.
G.f.: 1 + Sum_{k>0} 8 * x^(4*k) / (1 + x^(4*k))^2 + 4 * x^(2*k-1) / (1 - x^(2*k-1))^2 = 1 + Sum_{k>0} (2 + (-1)^k) * 4*k * x^(2*k) / (1 + x^(2*k)) + 4*(2*k - 1) * x^(2*k-1) / (1 - x^(2*k - 1)). - Michael Somos Oct 22 2005
a(2*n) = A000118(n). a(2*n + 1) = 4 * A008438(n). a(4*n) = A004011(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 2) = 8 * A008438(n). a(4*n + 3) = 16 * A097723(n). - Michael Somos Jul 05 2011
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EXAMPLE
| 1 + 4*q + 8*q^2 + 16*q^3 + 24*q^4 + 24*q^5 + 32*q^6 + 32*q^7 + 24*q^8 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}] (* Michael Somos Jul 05 2011 *)
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PROG
| (PARI) {a(n) = local(t); if( n<1, n>=0, t = 2^valuation( n, 2); 4 * sigma(n/t) * if( t>2, 6, t))} /* Michael Somos, Sep 17 2004 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A))^6 / (eta(x + A) * eta(x^8 + A))^4, n))} /* Michael Somos, Sep 17 2004 */
(PARI) {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0, 0; 0, 1, 0, 0; 0, 0, 2, 0; 0, 0, 0, 2], n)[n])} /* Michael Somos Oct 29 2005 */
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CROSSREFS
| Cf. A000118, A004011, A008438, A097723, A112610.
Sequence in context: A036302 A032377 A133690 * A160746 A160740 A181823
Adjacent sequences: A097054 A097055 A097056 * A097058 A097059 A097060
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 15 2004
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