

A281659


Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 16*y^2 + 24*z^2 + 48*w^2 a square, where x,y,z,w are nonnegative integers.


1



1, 2, 2, 2, 2, 1, 4, 1, 2, 3, 4, 1, 5, 1, 5, 1, 2, 2, 8, 1, 2, 5, 2, 2, 4, 3, 2, 8, 1, 5, 4, 2, 2, 3, 3, 4, 4, 3, 2, 2, 4, 2, 6, 1, 1, 3, 2, 3, 5, 2, 3, 7, 3, 1, 7, 1, 5, 2, 2, 4, 1, 1, 6, 7, 2, 1, 7, 2, 4, 5, 6, 0, 8, 3, 6, 8, 5, 3, 4, 2, 2
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OFFSET

0,2


COMMENTS

Conjecture: (i) a(n) > 0 except for n = 71, 85.
(ii) Any nonnegative integer n not among 39, 71 and 649 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 16*y^2 + 24*z^2 + 32*w^2 a square, where x,y,z,w are integers.
(iii) Each nonnegative integer n not among 5, 7, 23, 47, 93, 103, 109, 151, 191 and 911 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 3*y^2 + 8*z^2 + 16*w^2 a square, where x,y,z,w are integers.
(iv) Any nonnegative integer n not among 15, 19, 71, 103, 191 and 559 can be written as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 48*y^2 + 64*z^2 + 96*w^2 a square, where x,y,z,w are integers.
(v) Any nonnegative integer n not among 5, 93, 95, 161 and 309 can be written as x^2 + y^2 + z^2 + w^2 with 16*x^2 + 25*y^2 + 48*z^2 + 128*w^2 a square, where x,y,z,w are integers.
(vi) Each nonnegative integer not among 13, 19, 39, 41, 71, 109, 131, 193, 233, 377, 415 and 941 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 24*y^2 + 48*z^2 + 144*w^2 a square, where x,y,z,w are integers.
We have verified part (i) of the conjecture for n up to 1.2*10^5.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..10000
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.
ZhiWei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.


EXAMPLE

a(11) = 1 since 11 = 3^2 + 1^2 + 1^2 + 0^2 with 9*3^2 + 16*1^2 + 24*1^2 + 48*0 = 11^2.
a(170) = 1 since 170 = 3^2 + 6^2 + 2^2 + 11^2 with 9*3^2 + 16*6^2 + 24*2^2 + 48*11^2 = 81^2.
a(305) = 1 since 305 = 0^2 + 15^2 + 4^2 + 8^2 with 9*0^2 + 16*15^2 + 24*4^2 + 48*8^2 = 84^2.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[nx^2y^2z^2]&&SQ[9x^2+16y^2+24z^2+48*(nx^2y^2z^2)], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[nx^2]}, {z, 0, Sqrt[nx^2y^2]}]; Print[n, " ", r]; Label[aa]; Continue, {n, 0, 80}]


CROSSREFS

Cf. A000118, A000290, A271510, A271513, A271518.
Sequence in context: A231168 A276830 A064741 * A308619 A114294 A242050
Adjacent sequences: A281656 A281657 A281658 * A281660 A281661 A281662


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 26 2017


STATUS

approved



