A112607 Number of representations of n as a sum of a triangular number and twelve times a triangular number.

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

Author

James Sellers, Dec 21 2005

Keywords

• nonn

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).

Examples

			a(15) = 2 since we can write 15 = 15 + 12*0 = 3 + 12*1.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

A123484(24n+15) = 2*a(n). A112609(3n+4) = a(n).

Programs

Mathematica

a[n_] := DivisorSum[8n+13, KroneckerSymbol[-3, #]&]/2; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

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 */

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