A275656 - OEIS

%I #10 Aug 04 2016 06:39:17

%S 1,1,1,2,2,1,1,1,2,2,1,2,3,1,1,2,3,2,2,3,2,1,1,2,2,2,2,4,2,2,1,1,4,2,

%T 2,3,3,1,2,3,3,3,1,2,3,2,1,2,4,2,3,5,3,2,3,2,4,2,1,4,2,2,2,2,5,3,3,4,

%U 4,2,1,3,5,2,3,4,1,2,1,3

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

%C Conjecture: a(n) > 0 for all n > 0.

%C This is stronger than Lagrange's four-square theorem and the conjecture in A275648.

%H Zhi-Wei Sun, <a href="/A275656/b275656.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.

%e a(22) = 1 since 22 = 4^0*(1+1^2+2^2) + 4^2 with 1 < 2 < 4.

%e a(31) = 1 since 31 = 4^0*(1+1^2+2^2) + 5^2 with 1 < 2 < 5.

%e a(59) = 1 since 59 = 4^0*(1+0^2+3^2) + 7^2 with 0 < 3 < 7.

%e a(79) = 1 since 79 = 4^0*(1+2^2+5^2) + 7^2 with 2 < 5 < 7.

%e a(94) = 1 since 94 = 4^0*(1+2^2+5^2) + 8^2 with 2 < 5 < 8.

%e a(128) = 1 since 128 = 4^3*(1+0^2+0^2) + 8^2 with 0 = 0 < 8.

%e a(134) = 1 since 134 = 4^0*(1+4^2+6^2) + 9^2 with 4 < 6 < 9.

%e a(221) = 1 since 221 = 4*(1+3^2+5^2) + 9^2 with 3 < 5 < 9.

%e a(254) = 1 since 254 = 4^0*(1+3^2+10^2) + 12^2 with 3 < 10 < 12.

%e a(349) = 1 since 349 = 4*(1+5^2+7^2) + 7^2  with 5 < 7 = 7.

%e a(608) = 1 since 608 = 4^2*(1+0^2+1^2) + 24^2 with 0 < 1 < 24.

%e a(797) = 1 since 797 = 4*(1+0^2+4^2) + 27^2 with 0 < 4 < 27.

%e a(1181) = 1 since 1181 = 4*(1+9^2+9^2) + 23^2 with 9 = 9 < 23.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

%t Do[r=0;Do[If[SQ[n-4^k*(1+x^2+y^2)],r=r+1],{k,0,Log[4,n]},{x,0,Sqrt[(n-4^k)/(2*4^k+1)]},{y,x,Sqrt[(n-4^k*(1+x^2))/(4^k+1)]}];Print[n," ",r];Continue,{n,1,80}]

%Y Cf. A000118, A000290, A271518, A275648.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Aug 04 2016