OFFSET
0,2
COMMENTS
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
FORMULA
a(n) = d_{1, 3}(4n+3) - d_{2, 3}(4n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-3/4)*eta(q^2)^5*eta(q^12)^2/(eta(q)^2*eta(q^4)^2*eta(q^6)) in powers of q. - Michael Somos, May 20 2006
Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -2, 2, -1, 2, -3, 2, -2, ...]. - Michael Somos, May 20 2006
a(n)=A002324(4n+3). - Michael Somos, May 20 2006
Expansion of phi(q)*psi(q^6) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, May 20 2006, Sep 29 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 A164273. - Michael Somos, Aug 11 2009
a(3*n + 2) = 0. - Michael Somos, Aug 11 2009
EXAMPLE
a(22) = 4 since we can write 22 = 4^2 + 6*1 = (-4)^2 + 6*1 = 2^2 + 6*3 = (-2)^2 + 6*3.
G.f. = 1 + 2*x + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^15 + 2*x^16 + ... - Michael Somos, Aug 11 2009
G.f. = q^3 + 2*q^7 + 2*q^19 + q^27 + 2*q^31 + 2*q^39 + 2*q^43 + 2*q^63 + ... - Michael Somos, Aug 11 2009
MATHEMATICA
a[n_] := DivisorSum[4n+3, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)
PROG
(PARI) {a(n) = if(n<0, 0, sumdiv(4*n+3, d, kronecker(-3, d)))}; /* Michael Somos, May 20 2006 */
(PARI) {a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^12+A)^2 / eta(x+A)^2 / eta(x^4+A)^2 / eta(x^6+A), n))}; /* Michael Somos, May 20 2006 */
CROSSREFS
KEYWORD
nonn
AUTHOR
James A. Sellers, Dec 21 2005
STATUS
approved