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Search: id:A112606
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| A112606 |
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Number of representations of n as a sum of six times a square and a triangular number. |
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+0
7
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| 1, 1, 0, 1, 0, 0, 3, 2, 0, 2, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 3, 0, 0, 2, 2, 0, 4, 1, 0, 2, 0, 0, 0, 4, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 0, 2, 0, 0, 4, 2, 0, 0, 2, 0, 1, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 2, 4, 0, 4, 0, 0, 4, 0, 0
(list; graph; listen)
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OFFSET
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0,7
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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}(8n+1) - d_{2, 3}(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)^2*eta(q^12)^5/(eta(q)*eta(q^6)^2*eta(q^24)^2) in powers of q. - Michael Somos Sep 29 2006
Expansion of phi(q^6)*psi(q) in powers of q where phi(),psi() are Ramanujan theta functions.
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -4, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos Sep 29 2006
G.f.: (Sum_{k} x^(6k^2))(Sum_{k>0} x^((k^2-k)/2)). a(3n+2)=0. - Michael Somos Sep 29 2006
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EXAMPLE
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a(6) = 3 since we can write 6 = 6*1^2 + 0 = 6*(-1)^2 + 0 = 0 + 6
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, n=8*n+1; sumdiv(n, d, kronecker(-3, 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)^2*eta(x^12+A)^5/(eta(x+A)*eta(x^6+A)^2*eta(x^24+A)^2), n))} /* Michael Somos Sep 29 2006 */
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CROSSREFS
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A123484(24n+3)=A112604(2n)=A112608(3n)=a(n).
Sequence in context: A142071 A118972 A145878 this_sequence A108512 A054503 A122861
Adjacent sequences: A112603 A112604 A112605 this_sequence A112607 A112608 A112609
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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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