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A112607
Number of representations of n as a sum of a triangular number and twelve times a triangular number.
15
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
OFFSET
0,16
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) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
LINKS
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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
EXAMPLE
a(15) = 2 since we can write 15 = 15 + 12*0 = 3 + 12*1.
MATHEMATICA
a[n_] := DivisorSum[8n+13, KroneckerSymbol[-3, #]&]/2; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)
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 */
CROSSREFS
A123484(24n+15) = 2*a(n). A112609(3n+4) = a(n).
Sequence in context: A191269 A216602 A236233 * A161371 A327928 A364387
KEYWORD
nonn
AUTHOR
James A. Sellers, Dec 21 2005
STATUS
approved