login
A112606
Number of representations of n as a sum of six times a square and a triangular number.
13
1, 1, 0, 1, 0, 0, 3, 2, 0, 2, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 3, 0, 0, 2, 2, 0, 4, 1, 0, 2, 0, 0, 0, 4, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 0, 2, 0, 0, 4, 2, 0, 0, 2, 0, 1, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 2, 4, 0, 4, 0, 0, 4, 0, 0
OFFSET
0,7
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The greedy inverse starts 2, 0, 7, 6, 27, 300, 349, 14706, 216, 1035, 17107,... - R. J. Mathar, Apr 28 2020
REFERENCES
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
a(n) = d_{1, 3}(8n+1) - d_{2, 3}(8n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^12)^5 /(eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q. - Michael Somos, Sep 29 2006
Expansion of phi(q^6) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -4, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos, Sep 29 2006
G.f.: (Sum_{k} x^(6*k^2)) * (Sum_{k>0} x^((k^2-k)/2)). a(3*n+2)=0. - Michael Somos, Sep 29 2006
a(n) = A123484(24*n + 3) = A112604(2*n) = A112608(3*n). A131961(n) = a(3*n). A112608(n) = a(3*n + 1).
EXAMPLE
1 + x + x^3 + 3*x^6 + 2*x^7 + 2*x^9 + x^10 + 2*x^12 + x^15 + 2*x^16 + ...
q + q^9 + q^25 + 3*q^49 + 2*q^57 + 2*q^73 + q^81 + 2*q^97 + q^121 + 2*q^129 + ...
a(6) = 3 since we can write 6 = 6*1^2 + 0 = 6*(-1)^2 + 0 = 0 + 6.
MATHEMATICA
a[ n_] := If[ n < 0, 0, Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ 8 n + 1]}]] (* Michael Somos, Jun 16 2011 since V6 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 3, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]] (* Michael Somos, Jun 16 2011 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; sumdiv(n, d, kronecker(-3, d)))} /* 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^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))} /* Michael Somos, Sep 29 2006 */
CROSSREFS
KEYWORD
nonn
AUTHOR
James A. Sellers, Dec 21 2005
STATUS
approved