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Search: id:A097057
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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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+0
4
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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; listen)
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OFFSET
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0,2
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COMMENT
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One of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos Apr 01 2008
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REFERENCES
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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
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Y. Mimura, Almost Universal Quadratic Forms.
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FORMULA
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Euler transform of period 8 sequence [4, -2, 4, -8, 4, -2, 4, -4, ...]. - Michael Somos, Sep 17 2004
Multiplicative with a(n)=4b(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
G.f.: 1+Sum_{k>0} 8*x^(4k)/(1+x^(4k))^2 +4*x^(2k-1)/(1-x^(2k-1))^2 = 1+Sum_{k>0} (2+(-1)^k)4k x^(2k)/(1+x^(2k)) +4(2k-1)x^(2k-1)/(1-x^(2k-1)) . - Michael Somos Oct 22 2005
Expansion of (eta(q^2)eta(q^4))^6/(eta(q)eta(q^8))^4 in powers of q.
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.
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PROGRAM
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(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
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Sequence in context: A036302 A032377 A133690 this_sequence A160746 A160740 A046059
Adjacent sequences: A097054 A097055 A097056 this_sequence A097058 A097059 A097060
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Sep 15 2004
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