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A119395
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Number of nonnegative integer solutions to the equation x^2 + 3y^2 = n.
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9
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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
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MATHEMATICA
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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;
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PROG
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(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) }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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