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A282014 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 x and 121*x + 48*(y-z) are squares. 9
1, 2, 3, 3, 2, 2, 4, 2, 1, 5, 4, 3, 3, 2, 2, 3, 2, 4, 8, 4, 3, 3, 4, 2, 2, 6, 4, 7, 3, 1, 6, 1, 3, 7, 6, 5, 5, 3, 5, 4, 1, 4, 8, 5, 3, 4, 4, 2, 3, 5, 4, 9, 5, 3, 9, 4, 2, 7, 6, 2, 5, 2, 4, 4, 2, 5, 8, 8, 4, 4, 7, 2, 3, 6, 5, 9, 3, 2, 8, 2, 2 (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 = 8, 29, 31, 40, 94, 104, 143, 319, 671).

The author has proved that any nonnegative integer can be written as the sum of a fourth power and three squares.

We have verified a(n) > 0 for all n = 0..10^7.

See also A281976, A281977 and A282013 for similar conjectures.

Qing-Hu Hou at Tianjin University verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Jun 02 2019

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(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 0 = 0^2 and 121*0 + 48*(2-2) = 0^2.

a(29) = 1 since 29 = 0^2 + 5^2 + 2^2 + 0^2 with 0 = 0^2 and 121*0 + 48*(5-2) = 12^2.

a(31) = 1 since 31 = 1^2 + 2^2 + 1^2 + 5^2 with 1 = 1^2 and 121*1 + 48*(2-1) = 13^2.

a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 4 = 2^2 and 121*4 + 48*(2-2) = 22^2.

a(94) = 1 since 94 = 0^2 + 6^2 + 3^2 + 7^2 with 0 = 0^2 and 121*0 + 48*(6-3) = 12^2.

a(104) = 1 since 104 = 4^2 + 6^2 + 6^2 + 4^2 with 4 = 2^2 and 121*4 + 48*(6-6) = 22^2.

a(143) = 1 since 143 = 1^2 + 6^2 + 5^2 + 9^2 with 1 = 1^2 and 121*1 + 48*(6-5) = 13^2.

a(319) = 1 since 319 = 1^2 + 17^2 + 2^2 + 5^2 with 1 = 1^2 and 121*1 + 48*(17-2) = 29^2.

a(671) = 1 since 671 = 9^2 + 5^2 + 23^2 + 6^2 with 9 = 3^2 and 121*9 + 48*(5-23) = 15^2.

MATHEMATICA

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

Do[r=0; Do[If[SQ[n-x^4-y^2-z^2]&&SQ[121x^2+48(y-z)], r=r+1], {x, 0, n^(1/4)}, {y, 0, Sqrt[n-x^4]}, {z, 0, Sqrt[n-x^4-y^2]}]; Print[n, " ", r]; Continue, {n, 0, 80}]

CROSSREFS

Cf. A000118, A000290, A270969, A271518, A281939, A281941, A281975, A281976, A281977, A282013.

Sequence in context: A107901 A030423 A130631 * A241539 A213512 A218774

Adjacent sequences:  A282011 A282012 A282013 * A282015 A282016 A282017

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 04 2017

STATUS

approved

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Last modified November 17 00:08 EST 2019. Contains 329209 sequences. (Running on oeis4.)