0,4

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

a(n) = d_{1, 3}(4n+1) - d_{2, 3}(4n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.

Euler transform of period 12 sequence [0,1,2,-1,0,-2,0,-1,2,1,0,-2,...]. - Michael Somos Feb 14 2006

Number of representations of 2n as a sum of three times a triangular number and a triangular number.

Expansion of (psi(q)psi(q^3)+psi(-q)psi(-q^3))/2 in powers of q^2 where psi() is a Ramanujan theta function. - Michael Somos Feb 14 2006

G.f.: (Sum_{k} x^k^2)^3*(Sum_{k>0} x^((k^2-k)/2))^2 = Product_{k>0} (1-x^(4k))(1-x^(6k))(1+x^(2k))(1+x^(3k))^2/(1+x^(6k))^2 . - Michael Somos Feb 14 2006

a(12) = 3 since we can write 12 = 3(2)^2 + 0 = 3(-2)^2 + 0 = 0 + 2*6

2*12=24=3*1+21=3*3+15=3*6+6 so a(12)=3.

(PARI) a(n)=if(n<0, 0, n=4*n+1; sumdiv(n, d, (d%3==1)-(d%3==2)))

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^6+A)^5/eta(x^2+A)*(eta(x^4+A)/eta(x^3+A)/eta(x^12+A))^2, n))} /* Michael Somos Feb 14 2006 */

a(n)=A033762(2n)=A002324(4n+1).

Adjacent sequences: A112601 A112602 A112603 this_sequence A112605 A112606 A112607

Sequence in context: A060989 A135298 A006996 this_sequence A072627 A069848 A118682

nonn

James A. Sellers (sellersj(AT)math.psu.edu), Dec 21 2005