login
A112608
Number of representations of n as a sum of a twice a square and three times a triangular number.
15
1, 0, 2, 1, 0, 2, 0, 0, 2, 1, 0, 4, 0, 0, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 4, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 2, 2, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 4, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 2, 4, 0, 0
OFFSET
0,3
COMMENTS
The greedy inverse (first occurrence of n) starts 1, 0, 2, 18, 11, 900, 116, 44118, 515, 3105, 5702, ... - R. J. Mathar, Apr 28 2020
LINKS
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
FORMULA
a(n) = d_{1, 3}(8n+3) - d_{2, 3}(8n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Euler transform of period 24 sequence [0, 2, 1, -3, 0, 1, 0, -1, 1, 2, 0, -4, 0, 2, 1, -1, 0, 1, 0, -3, 1, 2, 0, -2, ...]. - Michael Somos, Jan 01 2006
Expansion of q^(-3/8)*(eta(q^4)^5*eta(q^6)^2)/(eta(q^2)^2*eta(q^3)*eta(q^8)^2) in powers of q.
a(n) = A002324(8*n+3).
EXAMPLE
a(11) = 4 since we can write 11 = 2*(2)^2 + 3*1 = 2*(-2)^2 + 3*1 = 2*(1)^2 + 3*3 = 2*(-1)^2 + 3*3
MATHEMATICA
eta[x_] := x^(1/24)*QPochhammer[x]; A112608[n_] := SeriesCoefficient[ q^(-3/8)*(eta[q^4]^5*eta[q^6]^2)/(eta[q^2]^2*eta[q^3]*eta[q^8]^2), {q, 0, n}]; Table[A112608[n], {n, 0, 50}] (* G. C. Greubel, Sep 25 2017 *)
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)^5*eta(x^6+A)^2/ eta(x^2+A)^2/eta(x^3+A)/eta(x^8)^2, n))} /* Michael Somos, Jan 01 2006 */
CROSSREFS
Sequence in context: A182033 A112214 A246962 * A058677 A262780 A033762
KEYWORD
nonn
AUTHOR
James A. Sellers, Dec 21 2005
STATUS
approved