

A275678


Number of ordered ways to write n as 4^k*(1+4*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x <= y.


5



1, 2, 1, 1, 3, 3, 1, 2, 3, 4, 2, 1, 2, 3, 2, 1, 4, 4, 1, 3, 5, 3, 1, 3, 5, 5, 3, 1, 2, 7, 2, 2, 5, 3, 3, 3, 6, 2, 2, 4, 6, 7, 1, 2, 4, 7, 1, 1, 3, 5, 5, 2, 5, 5, 4, 3, 8, 4, 2, 2, 1, 7, 3, 1, 6, 8, 2, 4, 8, 6, 2, 4, 6, 3, 4, 1, 3, 6, 2, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) Any positive integer can be written as 4^k*(1+4*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x <= z.
This is stronger than Lagrange's foursquare theorem. We have shown that each n = 1,2,3,... can be written as 4^k*(1+4*x^2+y^2) + z^2 with k,x,y,z nonnegative integers.
See also A275656, A275675 and A275676 for similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Refining Lagrange's foursquare theorem, arXiv:1604.06723 [math.GM], 2016.


EXAMPLE

a(12) = 1 since 12 = 4*(1+4*0^2+1^2) + 2^2 with 0 < 1.
a(19) = 1 since 19 = 4^0*(1+4*0^2+3^2) + 3^2 with 0 < 3.
a(61) = 1 since 61 = 4*(1+4*1^2+2^2) + 5^2 with 1 < 2.
a(125) = 1 since 125 = 4*(1+4*0^2+0^2) + 11^2 with 0 = 0.
a(359) = 1 since 359 = 4^0*(1+4*7^2+9^2) + 9^2 with 7 < 9.
a(196253) = 1 since 196253 = 4*(1+4*0^2+0^2) + 443^2 with 0 = 0.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n4^k*(1+4x^2+y^2)], r=r+1], {k, 0, Log[4, n]}, {x, 0, Sqrt[(n/4^k1)/5]}, {y, x, Sqrt[n/4^k14x^2]}]; Print[n, " ", r]; Continue, {n, 1, 80}]


CROSSREFS

Cf. A000118, A000290, A271518, A275648, A275656, A275675, A275676.
Sequence in context: A056670 A170925 A030189 * A273108 A306405 A114162
Adjacent sequences: A275675 A275676 A275677 * A275679 A275680 A275681


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Aug 05 2016


STATUS

approved



