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

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 A164272. - Michael Somos Aug 11 2009

a(3*n + 1) = 0. - Michael Somos Aug 11 2009

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.

1 + x^2 + 2*x^3 + 2*x^5 + x^6 + 2*x^9 + 3*x^12 + 2*x^14 + 2*x^15 + ... - Michael Somos Aug 11 2009

q + q^9 + 2*q^13 + 2*q^21 + q^25 + 2*q^37 + 3*q^49 + 2*q^57 + 2*q^61 + ... - Michael Somos Aug 11 2009

(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).

A112606(n) = a(2*n). 2 * A112607(n) = a(2*n + 1). A123884(n) = a(3*n). A112605(n) = a(3*n + 2). A131961(n) = a(6*n). A112608(n) =a(6*n + 2). 2 * A131963(n) = a(6*n + 3). 2 * A112609(n) = a(6*n + 5). - Michael Somos Aug 11 2009

Sequence in context: A060989 A135298 A006996 this_sequence A072627 A069848 A118682

Adjacent sequences: A112601 A112602 A112603 this_sequence A112605 A112606 A112607

nonn

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