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Search: id:A112603
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| A112603 |
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Number of representations of n as a sum of a square and a triangular number. |
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+0
4
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| 1, 3, 2, 1, 4, 2, 1, 4, 0, 2, 5, 2, 2, 0, 2, 3, 4, 2, 0, 6, 0, 1, 4, 0, 2, 4, 4, 0, 3, 2, 2, 4, 2, 0, 0, 2, 3, 8, 0, 2, 4, 0, 2, 0, 2, 3, 6, 0, 0, 4, 2, 2, 4, 2, 2, 3, 2, 2, 0, 4, 0, 4, 0, 0, 8, 2, 1, 4, 0, 0, 8, 2, 2, 0, 2, 2, 0, 2, 1, 4, 2, 4, 6, 0, 2, 4, 0, 4, 0, 0, 0, 7, 4, 0, 4, 2, 2, 0, 0, 0, 6, 2, 4, 4, 2
(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, 8}(8n+1) + d_{3, 8}(8n+1) - d_{5, 8}(8n+1) - d_{7, 8}(8n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-1/8)*eta(q^2)^7/(eta(q)^3*eta(q^4)^2) in powers of q. - Michael Somos Sep 29 2006
Expansion of phi(q)*psi(q) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos Sep 29 2006
Euler transform of period 4 sequence [ 3, -4, 3, -2, ...]. - Michael Somos Sep 29 2006
G.f.: (Sum_{k} x^(k^2))(Sum_{k>0} x^((k^2-k)/2)). - Michael Somos Sep 29 2006
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EXAMPLE
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a(4) = 4 since we can write 4 = 2^2 + 0 = (-2)^2 + 0 = 1^2 + 3 = (-1)^2 + 3
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, n=8*n+1; sumdiv(n, d, kronecker(-8, d)))} /* Michael Somos Sep 29 2006 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^7/(eta(x+A)^3*eta(x^4+A)^2), n))} /* Michael Somos Sep 29 2006 */
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CROSSREFS
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Adjacent sequences: A112600 A112601 A112602 this_sequence A112604 A112605 A112606
Sequence in context: A111572 A046900 A082727 this_sequence A097294 A060848 A006020
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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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