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A283196 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x + y and 2*x + z both squares, where x,y,z are integers with |y| <= |z|, and w is a positive integer. 6
1, 1, 1, 2, 2, 1, 3, 1, 1, 8, 1, 1, 6, 1, 3, 1, 3, 9, 2, 3, 3, 4, 4, 1, 7, 5, 2, 4, 3, 3, 6, 1, 5, 7, 1, 5, 4, 6, 4, 3, 2, 8, 3, 2, 11, 2, 6, 1, 6, 5, 1, 9, 4, 7, 11, 1, 3, 16, 1, 2, 5, 3, 14, 2, 7, 7, 4, 6, 3, 12, 6, 3, 8, 5, 2, 3, 5, 5, 9, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0.

(ii) Any positive integer n can be written as x^2 + y^2 + z^2 + w^2 such that both x + y and x + 2*z are squares, where x,y,z,w are integers with x >= 0 and w > 0.

(iii)  Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with 2*x + 2*y and 2*x + z both squares, where x,y,z,w are integers with x*y <= 0.

(iv) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with 2*|x-y| and 2*x + z both squares, where x,y,z,w are integers with x >= 0 and y >= 0.

By the linked JNT paper, any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that 2*x + y is a square, and also we can write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x - y (or 2*x - 2*y) is a square.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..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(2) = 1 since 2 = 0^2 + 0^2 + 1^2 + 1^2 with 2*0 + 0 = 0^2 and 2*0 + 1 = 1^2.

a(14) = 1 since 14 = 2^2 + 0^2 + (-3)^2 + 1^2 with 2*2 + 0 = 2^2 and 2*2 + (-3) = 1^2.

a(59) = 1 since 59 = 3^2 + 3^2 + (-5)^2 + 4^2 with 2*3 + 3 = 3^2 and 2*3 + (-5) = 1^2.

a(88) = 1 since 88 = (-2)^2 + 4^2 + 8^2 + 2^2 with 2*(-2) + 4 = 0^2 and 2*(-2) + 8 = 2^2.

a(131) = 1 since 131 = 0^2 + 1^2 + 9^2 + 7^2 with 2*0 + 1 = 1^2 and 2*0 + 9 = 3^2.

a(219) = 1 since 219 = 8^2 + (-7)^2 + 9^2 + 5^2 with 2*8 + (-7) = 3^2 and 2*8 + 9 = 5^2.

a(249) = 1 since 249 = (-4)^2 + 8^2 + 12^2 + 5^2 with 2*(-4) + 8 = 0^2 and 2*(-4) + 12 = 2^2.

a(312) = 1 since 312 = 6^2 + 4^2 + (-8)^2 + 14^2 with 2*6 + 4 = 4^2 and 2*6 + (-8) = 2^2.

a(323) = 1 since 323 = 9^2 + 7^2 + 7^2 + 12^2 with 2*9 + 7 = 5^2.

a(536) = 1 since 536 = (-6)^2 + 12^2 + 16^2 + 10^2 with 2*(-6) + 12 = 0^2 and 2*(-6) + 16 = 2^2.

a(888) = 1 since 888 = 14^2 + 8^2 + (-12)^2 + 22^2 with 2*14 + 8 = 6^2 and 2*14 + (-12) = 4^2.

a(1464) = 1 since 1464 = 2^2 + 0^2 + (-4)^2 + 38^2 with 2*2 + 0 = 2^2 and 2*2 + (-4) = 0^2.

a(4152) = 1 since 4152 = 30^2 + 4^2 + (-56)^2 + 10^2 with 2*30 + 4 = 8^2 and 2*30 + (-56) = 2^2.

MATHEMATICA

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

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

CROSSREFS

Cf. A000118, A000290, A271518, A281975, A281976, A281939, A283170.

Sequence in context: A051135 A325541 A260258 * A238882 A279287 A135352

Adjacent sequences:  A283193 A283194 A283195 * A283197 A283198 A283199

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 02 2017

STATUS

approved

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Last modified December 3 15:06 EST 2021. Contains 349463 sequences. (Running on oeis4.)