OFFSET
1,3
COMMENTS
This is motivated by the author's 1-3-5-Conjecture which states that any nonnegative integer can be expressed as the sum of a square and a term of the current sequence.
Clearly, any term times a fourth power is also a term of this sequence. By the Gauss-Legendre theorem on sums of three squares, no term has the form 4^k*(8m+7) with k and m nonnegative integers.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
EXAMPLE
a(4) = 3 since 3 = 1^2 + 1^2 + 1^2 with 1 + 3*1 + 5*1 = 3^2.
a(5) = 8 since 8 = 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0; Do[Do[If[SQ[m-x^2-y^2]&&SQ[x+3y+5*Sqrt[m-x^2-y^2]], n=n+1; Print[n, " ", m]; Goto[aa]], {x, 0, Sqrt[m]}, {y, 0, Sqrt[m-x^2]}]; Label[aa]; Continue, {m, 0, 146}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 23 2016
STATUS
approved