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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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15
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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
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OFFSET
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0,16
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COMMENTS
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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) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
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FORMULA
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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
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EXAMPLE
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a(15) = 2 since we can write 15 = 15 + 12*0 = 3 + 12*1.
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MATHEMATICA
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a[n_] := DivisorSum[8n+13, KroneckerSymbol[-3, #]&]/2; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)
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PROG
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(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
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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