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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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12
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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; refs; listen; history; internal format)
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OFFSET
| 0,2
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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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LINKS
| M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| 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
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A164273. - Michael Somos Aug 11 2009
a(3*n + 2) = 0. - Michael Somos Aug 11 2009
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EXAMPLE
| a(22) = 4 since we can write 22 = 4^2 + 6*1 = (-4)^2 + 6*1 = 2^2 + 6*3 = (-2)^2+ 6*3
1 + 2*x + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^15 + 2*x^16 + ... - Michael Somos Aug 11 2009
q^3 + 2*q^7 + 2*q^19 + q^27 + 2*q^31 + 2*q^39 + 2*q^43 + 2*q^63 + ... - Michael Somos Aug 11 2009
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PROG
| (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
| A112608(n) = a(2*n). 2 * A112609(n) = a(2*n + 1). A112604(n) = a(3*n). 2 * A121361(n) = a(3*n + 1). A112606(n) = a(6*n). 2 * A131962(n) = a(6*n + 1). 2 * A112607(n) = a(6*n + 3). 2 * A131964(n) = a(6*n + 4). - Michael Somos Aug 11 2009
Sequence in context: A118683 A175800 A161116 * A111775 A025844 A035461
Adjacent sequences: A112602 A112603 A112604 * A112606 A112607 A112608
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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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