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A282561 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2*x - y and 4*z^2 + 724*z*w + w^2 are squares. 4
1, 2, 2, 2, 4, 3, 5, 1, 3, 4, 2, 2, 2, 5, 7, 3, 2, 5, 3, 1, 4, 7, 7, 3, 2, 2, 2, 4, 3, 8, 8, 3, 2, 2, 4, 4, 9, 3, 9, 3, 4, 5, 6, 3, 3, 7, 5, 2, 2, 11, 6, 5, 4, 7, 7, 4, 2, 4, 3, 2, 2, 5, 10, 6, 4, 5, 9, 1, 7, 8, 10, 4, 4, 5, 6, 5, 3, 9, 3, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 7, 19, 67, 191, 235, 265, 347, 888, 2559).

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 nonnegative integers and z*(z-w) = 0. Whether z = 0 or z = w, the number 4*z^2 + 724*z*w + w^2 is definitely a square.

See also A282562 for a similar conjecture.

LINKS

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

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

EXAMPLE

a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with 2*1 - 2 = 0^2 and 4*1^2 + 724*1*1 + 1^2 = 27^2.

a(19) = 1 since 19 = 1^2 + 1^2 + 1^2 + 4^2 with 2*1 - 1 = 1^2 and 4*1^2 + 724*1*4 + 4^2 = 54^2.

a(67) = 1 since 67 = 4^2 + 7^2 + 1^2 + 1^2 with 2*4 - 7 = 1^2 and 4*1^2 + 724*1*1 + 1^2 = 27^2.

a(191) = 1 since 191 = 9^2 + 2^2 + 5^2 + 9^2 with 2*9 - 2 = 4^2 and 4*5^2 + 724*5*9 + 9^2 = 181^2.

a(235) = 1 since 235 = 7^2 + 13^2 + 1^2 + 4^2 with 2*7 - 13 = 1^2 and 4*1^2 + 724*1*4 + 4^2 = 54^2.

a(265) = 1 since 265 = 4^2 + 7^2 + 10^2 + 10^2 with 2*4 - 7 = 1 and 4*10^2 + 724*10*10 + 10^2 = 270^2.

a(347) = 1 since 347 = 8^2 + 7^2 + 15^2 + 3^2 with 2*8 - 7 = 3^2 and 4*15^2 + 724*15*3 + 3^2 = 183^2.

a(888) = 1 since 888 = 14^2 + 12^2 + 8^2 + 22^2 with 2*14 - 12 = 4^2 and 4*8^2 + 724*8*22 + 22^2 = 358^2.

a(2559) = 1 since 2559 = 26^2 + 3^2 + 5^2 + 43^2 with 2*26 - 3 = 7^2 and 4*5^2 + 724*5*43 + 43^2 = 397^2.

MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

Do[r=0; Do[If[SQ[2x-y], Do[If[SQ[n-x^2-y^2-z^2]&&SQ[4z^2+724z*Sqrt[n-x^2-y^2-z^2]+(n-x^2-y^2-z^2)], r=r+1], {z, 0, Sqrt[n-x^2-y^2]}]], {y, 0, Sqrt[4n/5]}, {x, Ceiling[y/2], Sqrt[n-y^2]}]; Print[n, " ", r]; Continue, {n, 0, 80}]

CROSSREFS

Cf. A000118, A000290, A271518, A281976, A282463, A282494, A282495, A282545, A282562.

Sequence in context: A247301 A239858 A031437 * A237598 A138241 A234615

Adjacent sequences:  A282558 A282559 A282560 * A282562 A282563 A282564

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 18 2017

STATUS

approved

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Last modified February 16 15:17 EST 2020. Contains 331961 sequences. (Running on oeis4.)