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A281975 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that both x and |x-y| are squares. 7
1, 3, 4, 3, 2, 4, 4, 2, 1, 4, 5, 3, 1, 2, 3, 2, 3, 5, 7, 3, 4, 5, 2, 1, 2, 6, 9, 6, 2, 4, 6, 3, 4, 6, 8, 4, 6, 5, 4, 3, 2, 11, 10, 4, 1, 7, 5, 1, 3, 3, 11, 9, 7, 5, 6, 4, 2, 7, 5, 4, 2, 6, 4, 3, 2, 7, 15, 4, 4, 6, 5, 1, 2, 6, 7, 7, 3, 6, 4, 2, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Conjecture: (i) a(n) > 0 for all n = 0,1,2,....

(ii) Each nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that |x-y| and 2*(y-z) (or 2*(z-y)) are both squares.

(iii) For each ordered pair (a,b) = (2,1), (3,1), (9,5), (14,10), any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x and |a*x-b*y| are both squares.

The author has proved that each 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 (or x-y, or 2(x-y)) is a square.

See also A281976 and A281977 for similar conjectures.

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.

EXAMPLE

a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and |0-0| = 0^2.

a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 = 1^2 and |1-1| = 0^2.

a(44) = 1 since 44 = 1^2 + 5^2 + 3^2 + 3^2 with 1 = 1^2 and |1-5| = 2^2.

a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 = 1^2 and |1-1| = 0^2.

a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 = 1^2 and |1-5| = 2^2.

a(95) = 1 since 95 = 1^2 + 2^2 + 3^2 + 9^2 with 1 = 1^2 and |1-2| = 1^2.

a(140) = 1 since 140 = 9^2 + 5^2 + 3^2 + 5^2 with 9 = 3^2 and |9-5| = 2^2.

a(428) = 1 since 428 = 9^2 + 13^2 + 3^2 + 13^2 with 9 = 3^2 and |9-13| = 2^2.

a(568) = 1 since 568 = 4^2 + 8^2 + 2^2 + 22^2 with 4 = 2^2 and |4-8| = 2^2.

a(632) = 1 since 632 = 16^2 + 12^2 + 6^2 + 14^2 with 16 = 4^2 and |16-12| = 2^2.

a(1144) = 1 since 1144 = 16^2 + 20^2 + 2^2 + 22^2 with 16 = 4^2 and |16-20| = 2^2.

a(1544) = 1 since 1544 = 0^2 + 0^2 + 10^2 + 38^2 with 0 = 0^2 and |0-0| = 0^2.

MATHEMATICA

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

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

CROSSREFS

Cf. A000118, A000290, A270969, A271775, A281939, A281941, A281976, A281977.

Sequence in context: A258451 A164358 A275638 * A133617 A199286 A188722

Adjacent sequences:  A281972 A281973 A281974 * A281976 A281977 A281978

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 03 2017

STATUS

approved

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Last modified February 16 20:45 EST 2019. Contains 320189 sequences. (Running on oeis4.)