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A323381 revision #3

A323381

Number of solutions to n = x^2 + 2*y^2 + 5*z^2 + x*z in integers.

1, 2, 2, 4, 2, 4, 4, 12, 2, 14, 0, 8, 4, 16, 0, 8, 2, 8, 6, 22, 4, 4, 12, 12, 4, 30, 0, 20, 12, 8, 8, 8, 2, 16, 4, 16, 14, 28, 12, 8, 0, 12, 0, 40, 8, 12, 16, 12, 4, 22, 2, 28, 16, 24, 8, 32, 0, 48, 0, 4, 8, 20, 8, 36, 2, 16, 8, 36, 8, 12, 16, 8, 6, 32, 0, 28

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OFFSET

0,2

COMMENTS

If n<0, then a(n)=0. If n>0, then a(n) is even since (-x, -y, -z) is a solution if (x, y, z) is.

Rouse [2014] conjectures that the ternary quadratic form x^2 + 2y^2 + 5z^2 + xz represents all positive odd integers.

LINKS

Table of n, a(n) for n=0..75.

J. Rouse, Quadratic forms representing all odd positive integers Amer. J. Math, 136 (2014), no. 6, 1693-1745.

K. S. Williams, Eveything You Wanted to Know about ax^2+by^2+cz^2+dt^2 But Were Afraid To Ask, Amer. Math. Monthly, Vol. 125, No. 9, (2018), 797-810. See page 803.

EXAMPLE

G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 12*x^7 + ...

MATHEMATICA

a[ n_] := Length @ FindInstance[ x^2 + 2 y^2 + 5 z^2 + x z == n, {x, y, z}, Integers, 10^8];

PROG

(PARI) {a(n) = if( n<1, n==0, 2*qfrep([4, 0, 0; 0, 2, 1; 0, 1, 10], 2*n)[2*n])};

CROSSREFS

Sequence in context: A116466 A116467 A079314 * A060609 A330882 A205138

Adjacent sequences: A323378 A323379 A323380 * A323382 A323383 A323384

KEYWORD

nonn

AUTHOR

Michael Somos, Jan 12 2019

STATUS

approved