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A112607
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Number of representations of n as a sum of a triangular number and twelve times a triangular number.
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7
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1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,16
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| a(n) = 1/2*( d_{1, 3}(8n+13) - d_{2, 3}(8n+13) ) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-13/8)*(eta(q^2)*eta(q^24))^2/(eta(q)*eta(q^12)) in powers of q. - Michael Somos Sep 29 2006
Expansion of psi(q)*psi(q^12) in powers of q where psi() is a Ramanujan theta function. - Michael Somos Sep 29 2006
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos Sep 29 2006
a(3n+2)=0. - Michael Somos Sep 29 2006
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EXAMPLE
| a(15) = 2 since we can write 15 = 15 + 12*0 = 3 + 12*1
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PROG
| (PARI) {a(n)=if(n<0, 0, n=8*n+13; sumdiv(n, d, kronecker(-3, d))/2)} /* 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^24+A)^2/eta(x+A)/eta(x^12+A), n))} /* Michael Somos Sep 29 2006 */
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CROSSREFS
| A123484(24n+15)=2*a(n). A112609(3n+4)=a(n).
Sequence in context: A078595 A078128 A191269 * A161371 A147645 A091970
Adjacent sequences: A112604 A112605 A112606 * A112608 A112609 A112610
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KEYWORD
| nonn
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AUTHOR
| James A. Sellers (sellersj(AT)math.psu.edu), Dec 21 2005
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