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A283170
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Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 2*y and z + 2*w both squares, where x and w are nonnegative integers, and y and z are integers.
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8
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1, 2, 2, 1, 2, 6, 5, 2, 1, 4, 5, 1, 1, 2, 3, 2, 2, 6, 9, 3, 5, 8, 7, 6, 2, 4, 2, 2, 1, 5, 7, 6, 2, 8, 10, 3, 5, 5, 4, 4, 1, 1, 8, 1, 2, 6, 7, 4, 1, 7, 12, 3, 5, 5, 7, 10, 2, 7, 9, 1, 2, 8, 8, 12, 2, 11, 13, 2, 7, 11, 9, 8, 3, 4, 10, 3, 2, 6, 6, 10, 6
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OFFSET
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0,2
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COMMENTS
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Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
(ii) Any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x + 3*y and z + 3*w both squares, where x,y,z are integers and w is a nonnegative integer.
(iii) Every nonnegative integer can be expressed as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x + 2*y and z + 3*w are squares.
(vi) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w an integer such that |2*x-y| is a square and |2*z-w| is twice a square. Also, each nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w an integer such that |2*x-y| is twice a square and |2*z-w| is a square.
(v) Every n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that |(x-2*y)*(z-2*w)| is twice a square. Also, any positive integer n can be written as x^2 + y^2 + z^2 + w^2 with x a positive integer and y,z,w nonnegative integers such that (2*x+y)*(2*z-w) is twice a square.
(vi) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x + 2*y and z^2 - w^2 (or z^2 + 8*w^2, or 7*z^2 + 9*w^2) are squares.
(vii) Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2*x - y and 64*z^2 - 84*z*w + 21*w^2 (or 81*z^2 - 112*z*w + 56*w^2) are squares.
By the linked JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + 2*y is a square.
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LINKS
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FORMULA
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a(3) = 1 since 3 = 1^2 + 0^2 + (-1)^2 + 1^2 with 1 + 2*0 = 1^2 and (-1)+2*1 = 1^2.
a(4) = 2 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 + 2*0 = 0^2 and 0 + 2*2 = 2^2, and also 4 = 0^2 + 2^2 + 0^2 + 0^2 with 0 + 2*2 = 2^2 and 0 + 2*0 = 0^2.
a(8) = 1 since 8 = 0^2 + 2^2 + 0^2 + 2^2 with 0 + 2*2 = 2^2 and 0 + 2*2 = 2^2.
a(11) = 1 since 11 = 3^2 + (-1)^2 + 1^2 + 0^2 with 3 + 2*(-1) = 1^2 and 1 + 2*0 = 1^2.
a(12) = 1 since 12 = 3^2 + (-1)^2 + (-1)^2 + 1^2 with 3 + 2*(-1) = 1^2 and (-1) + 2*1 = 1^2.
a(28) = 1 since 28 = 3^2 + (-1)^2 + 3^2 + 3^2 with 3 + 2*(-1) = 1^2 and 3 + 2*3 = 3^2.
a(40) = 1 since 40 = 4^2 + (-2)^2 + (-4)^2 + 2^2 with 4 + 2*(-2) = 0^2 and (-4) + 2*2 = 0^2.
a(41) = 1 since 41 = 6^2 + (-1)^2 + 0^2 + 2^2 with 6 + 2*(-1) = 2^2 and 0 + 2*2 = 2^2.
a(332) = 1 since 332 = 11^2 + 7^2 + (-9)^2 + 9^2 with 11 + 2*7 = 5^2 and (-9) + 2*9 = 3^2.
a(443) = 1 since 443 = 19^2 + (-9)^2 + 1^2 + 0^2 with 19 + 2*(-9) = 1^2 and 1 + 2*0 = 1^2.
a(488) = 1 since 488 = 12^2 + 2^2 + (-12)^2 + 14^2 with 12 + 2*2 = 4^2 and (-12) + 2*14 = 4^2.
a(808) = 1 since 808 = 8^2 + 14^2 + (-8)^2 + 22^2 with 8 + 2*14 = 6^2 and (-8) + 2*22 = 6^2.
a(892) = 1 since 892 = 27^2 + (-1)^2 + (-9)^2 + 9^2 with 27 + 2*(-1) = 5^2 and (-9) + 2*9 = 3^2.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[x+2(-1)^j*y], Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(-1)^k*z+2*Sqrt[n-x^2-y^2-z^2]], r=r+1], {z, 0, Sqrt[n-x^2-y^2]}, {k, 0, Min[z, 1]}]], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {j, 0, Min[y, 1]}]; Print[n, " ", r]; Continue, {n, 0, 80}]
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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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