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 A119395 Number of nonnegative integer solutions to the equation x^2 + 3y^2 = n. 8
 1, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 3, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 3, 0, 0, 1, 0, 1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The number of integer solutions is given by A033716. Records 1, 2, 3, 5, 6, 9, 12, 14, 18, ... occur at 0, 4, 28, 196, 364, 2548, 6916, 33124, 48412, ... - Antti Karttunen, Nov 20 2017 LINKS Antti Karttunen, Table of n, a(n) for n = 0..65537 N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references) FORMULA For n > 0, a(n) = (A033716(n) + 2)/4 if n is a square or a triple of a square; otherwise a(n) = A033716(n)/4. Alternatively, a(n) = ceiling(A033716(n)/4). G.f.: (1 + theta_3(q))*(1 + theta_3(q^3))/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018 MATHEMATICA QP = QPochhammer; s = (QP[q^2]*QP[q^6])^5/(QP[q]*QP[q^3]*QP[q^4]*QP[q^12])^2 + O[q]^105; A033716 = CoefficientList[s, q]; A119395 = Ceiling[A033716/4] (* Jean-François Alcover, Jul 02 2018 *) PROG (PARI) { A033716(n) = local(f, B); f=factorint(n); B=1; for(i=1, matsize(f)[1], if(f[i, 1]%3==1, B*=f[i, 2]+1); if(f[i, 1]%3==2, if(f[i, 2]%2, return(0)))); if(n%4, 2*B, 6*B) } { a(n) = ceil(A033716(n)/4) } CROSSREFS Cf. A033716, A096936. Sequence in context: A005089 A364127 A340999 * A087476 A307505 A035162 Adjacent sequences: A119392 A119393 A119394 * A119396 A119397 A119398 KEYWORD nonn AUTHOR Max Alekseyev, May 16 2006 STATUS approved

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Last modified May 18 15:59 EDT 2024. Contains 372664 sequences. (Running on oeis4.)