A273021 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x*y + y*z - z*w - w*x a square, where w is a positive integer and x,y,z are nonnegative integers with x <= y.

1, 1, 2, 2, 2, 3, 2, 1, 2, 2, 1, 3, 3, 4, 2, 2, 3, 5, 2, 2, 4, 1, 1, 3, 3, 4, 7, 4, 4, 1, 1, 1, 4, 4, 2, 4, 4, 6, 5, 2, 5, 7, 3, 3, 3, 4, 1, 3, 5, 4, 5, 6, 2, 8, 1, 4, 4, 4, 3, 2, 5, 5, 4, 2, 5, 7, 2, 3, 4, 5, 1, 5, 4, 5, 6, 5, 3, 4, 3, 2

Offset

1,3

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 11, 31, 47, 55, 71, 105, 115, 119, 253, 383, 385, 4^k*m (k = 0,1,2,... and m = 2, 22, 23, 30, 330).

(ii) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with (x+y)*(z+w) a square, where w is an integer and x,y,z are nonnegative integers with x <= y >= z >= |w|.

See arXiv:1604.06723 for more conjectural refinements of Lagrange's four-square theorem.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Example

a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 = 0 and 2*0*0 + 0*0 - 0*1 - 1*0 = 0^2.
a(2) = 1 since 2 = 0^2 + 1^2 + 0^2 + 1^2 with 0 < 1 and 2*0*1 + 1*0 - 0*1 - 1*0 = 0^2.
a(11) = 1 since 11 = 0^2 + 1^2 + 3^2 + 1^2 with 0 < 1 and 2*0*1 + 1*3 - 3*1 - 1*0 = 0^2.
a(22) = 1 since 22 = 0^2 + 3^2 + 2^2 + 3^2 with 0 < 3 and 2*0*3 + 3*2 - 2*3 - 3*0 = 0^2.
a(23) = 1 since 23 = 2^2 + 3^2 + 3^2 + 1^2 with 2 < 3 and 2*2*3 + 3*3 - 3*1 - 1*2 = 4^2.
a(30) = 1 since 30 = 1^2 + 3^2 + 2^2 + 4^2 with 1 < 3 and 2*1*3 + 3*2 - 2*4 - 4*1 = 0^2.
a(31) = 1 since 31 = 3^2 + 3^2 + 2^2 + 3^2 with 3 = 3 and
2*3*3 + 3*2 - 2*3 -3*3 = 3^2.
a(47) = 1 since 47 = 3^2 + 5^2 + 2^2 + 3^2 with 3 < 5 and 2*3*5 + 5*2 - 2*3 - 3*3 = 5^2.
a(55) = 1 since 55 = 1^2 + 7^2 + 2^2 + 1^2 with 1 < 7 and 2*1*7 + 7*2 - 2*1 - 1*1 = 5^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 < 5 and 2*1*5 + 5*3 - 3*6 - 6*1 = 1^2.
a(105) = 1 since 105 = 1^2 + 6^2 + 2^2 + 8^2 with 1 < 6 and 2*1*6 + 6*2 - 2*8 - 8*1 = 0^2.
a(115) = 1 since 115 = 1^2 + 8^2 + 7^2 + 1^2 with 1 < 8 and 2*1*8 + 8*7 - 7*1 - 1*1 = 8^2.
a(119) = 1 since 119 = 1^2 + 6^2 + 1^2 + 9^2 with 1 < 6 and 2*1*6 + 6*1 - 1*9 - 9*1 = 0^2.
a(253) = 1 since 253 = 2^2 + 8^2 + 11^2 + 8^2 with 2 < 8 and 2*2*8 + 8*11 - 11*8 - 8*2 = 4^2.
a(330) = 1 since 330 = 4^2 + 13^2 + 8^2 + 9^2 with 4 < 13 and 2*4*13 + 13*8 - 8*9 - 9*4 = 10^2.
a(383) = 1 since 383 = 9^2 + 14^2 + 5^2 + 9^2 with 9 < 14 and 2*9*14 + 14*5 - 5*9 - 9*9 = 14^2.
a(385) = 1 since 385 = 4^2 + 12^2 + 0^2 + 15^2 with 4 < 12 and 2*4*12 + 12*0 - 0*15 - 15*4 = 6^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[y*(2x+z)-Sqrt[n-x^2-y^2-z^2]*(x+z)], r=r+1], {x, 0, Sqrt[(n-1)/2]}, {y, x, Sqrt[n-1-x^2]}, {z, 0, Sqrt[n-1-x^2-y^2]}]; Print[n, " ", r]; Continue, {n, 1, 80}]

Crossrefs

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977.

Sequence in context: A116987 A219838 A159800 * A365722 A071137 A333253

Adjacent sequences: A273018 A273019 A273020 * A273022 A273023 A273024

Keyword

nonn

Author

Zhi-Wei Sun, May 13 2016

Status

approved