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 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 (list; graph; refs; listen; history; text; internal format)
 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 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 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 Adjacent sequences:  A112605 A112606 A112607 * A112609 A112610 A112611 KEYWORD nonn AUTHOR James A. Sellers, Dec 21 2005 STATUS approved

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Last modified June 19 14:40 EDT 2021. Contains 345140 sequences. (Running on oeis4.)