A273278 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^2*y^2 + 3*y^2*z^2 + 2*z^2*w^2 is a square.

1, 4, 4, 2, 4, 8, 4, 1, 4, 8, 8, 4, 2, 9, 6, 3, 4, 12, 9, 6, 8, 8, 5, 2, 4, 14, 15, 6, 1, 14, 9, 2, 4, 9, 12, 8, 8, 9, 11, 1, 8, 18, 7, 4, 4, 17, 8, 3, 2, 12, 18, 9, 9, 17, 15, 4, 6, 8, 8, 10, 3, 15, 13, 5, 4, 22, 15, 6, 12, 15, 13

Offset

0,2

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 4^k*m (k = 0,1,2,... and m = 7, 39, 87, 183, 231, 807, 879, 959, 1479, 2391, 2519, 2759, 4359, 10887).

See part (ii) of the conjecture in A269400 for similar conjectures.

For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

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

Example

a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2^2*1^2 + 3*1^2*1^2 + 2*1^2*1^2 = 3^2.
a(39) = 1 since 39 = 2^2 + 1^2 + 5^2 + 3^2 with 2^2*1^2 + 3*1^2*5^2 + 2*5^2*3^2 = 23^2.
a(87) = 1 since 87 = 2^2 + 1^2 + 1^2 + 9^2 with 2^2*1^2 + 3*1^2*1^2 + 2*1^2*9^2 = 13^2.
a(183) = 1 since 183 = 10^2 + 7^2 + 5^2 + 3^2 with 10^2*7^2 + 3*7^2*5^2 + 2*5^2*3^2 = 95^2.
a(231) = 1 since 231 = 10^2 + 1^2 + 9^2 + 7^2 with 10^2*1^2 + 3*1^2*9^2 + 2*9^2*7^2 = 91^2.
a(807) = 1 since 807 = 10^2 + 23^2 + 3^2 + 13^2 with 10^2*23^2 + 3*23^2*3^2 + 2*3^2*13^2 = 265^2.
a(879) = 1 since 879 = 14^2 + 11^2 + 21^2 + 11^2 with 14^2*11^2 + 3*11^2*21^2 + 2*21^2*11^2 = 539^2.
a(959) = 1 since 959 = 10^2 + 15^2 + 25^2 + 3^2 with 10^2*15^2 + 3*15^2*25^2 + 2*25^2*3^2 = 675^2.
a(1479) = 1 since 1479 = 34^2 + 11^2 + 11^2 + 9^2 with 34^2*11^2 + 3*11^2*11^2 + 2*11^2*9^2 = 451^2.
a(2391) = 1 since 2391 = 34^2 + 11^2 + 5^2 + 33^2 with 34^2*11^2 + 3*11^2*5^2 + 2*5^2*33^2 = 451^2.
a(2519) = 1 since 2519 = 42^2 + 1^2 + 27^2 + 5^2 with 42^2*1^2 + 3*1^2*27^2 + 2*27^2*5^2 = 201^2.
a(2759) = 1 since 2759 = 26^2 + 21^2 + 11^2 + 39^2 with 26^2*21^2 + 3*21^2*11^2 + 2*11^2*39^2 = 909^2.
a(4359) = 1 since 4359 = 46^2 + 19^2 + 19^2 + 39^2 with 46^2*19^2 + 3*19^2*19^2 + 2*19^2*39^2 = 1501^2.
a(10887) = 1 since 10887 = 31^2 + 85^2 + 51^2 + 10^2 with 31^2*85^2 + 3*85^2*51^2 + 2*51^2*10^2 = 7990^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x^2*y^2+3*y^2*z^2+2z^2*(n-x^2-y^2-z^2)], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[n-x^2-y^2]}]; Print[n, " ", r]; Continue, {n, 0, 70}]

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, A273021, A273107, A273108, A273110, A273134.

Sequence in context: A112108 A373822 A021230 * A375850 A213089 A246644

Adjacent sequences: A273275 A273276 A273277 * A273279 A273280 A273281

Keyword

nonn

Author

Zhi-Wei Sun, May 18 2016

Status

approved