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Search: id:A112605
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| A112605 |
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Number of representations of n as a sum of a square and six times a triangular number. |
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+0
7
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| 1, 2, 0, 0, 2, 0, 1, 2, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 1, 2, 0, 0, 4, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 3, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 1, 4, 0, 0, 4, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 4, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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FORMULA
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a(n) = d_{1, 3}(4n+3) - d_{2, 3}(4n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-3/4)eta(q^2)^5*eta(q^12)^2/(eta(q)^2*eta(q^4)^2*eta(q^6)) in powers of q. - Michael Somos May 20 2006
Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -2, 2, -1, 2, -3, 2, -2, ...]. - Michael Somos May 20 2006
a(n)=A002324(4n+3). - Michael Somos May 20 2006
Expansion of phi(q)*psi(q^6) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos May 20 2006, Sep 29 2006
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EXAMPLE
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a(22) = 4 since we can write 22 = 4^2 + 6*1 = (-4)^2 + 6*1 = 2^2 + 6*3 = (-2)^2+ 6*3
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, sumdiv(4*n+3, d, kronecker(-3, d)))} /* Michael Somos May 20 2006 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^12+A)^2/ eta(x+A)^2/eta(x^4+A)^2/eta(x^6+A), n))} /* Michael Somos May 20 2006 */
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CROSSREFS
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Adjacent sequences: A112602 A112603 A112604 this_sequence A112606 A112607 A112608
Sequence in context: A047919 A101670 A118683 this_sequence A111775 A025844 A035461
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KEYWORD
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nonn
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AUTHOR
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James A. Sellers (sellersj(AT)math.psu.edu), Dec 21 2005
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