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%I #18 Mar 06 2018 03:06:16
%S 1,1,2,1,3,2,1,1,4,3,3,2,3,1,3,1,3,4,4,3,5,3,3,2,4,3,3,1,5,3,3,1,6,3,
%T 4,4,5,4,4,3,6,5,4,3,5,3,4,2,5,4,5,3,4,3,5,1,5,5,3,3,3,3,5,1,5,6,3,3,
%U 6,4,4,4,6,5,5,4,6,4,5,3
%N Number of the integers 4^k*(4*u(m)^2+1) (k,m = 0,1,2,...) such that n^2 - 4^k*(4*u(m)^2+1) can be written as the sum of two squares, where u(0) = 0, u(1) = 1, and u(j+1) = 4*u(j) - u(j-1) for j = 1,2,3,....
%C Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 2^k*m (k = 0,1,2,... and m = 1, 7).
%C This curious conjecture indicates that any positive square can be written as (2^k)^2 + (2^(k+1)*u(m))^2 + x^2 + y^2 with k,m,x,y nonnegative integers. In the 2017 JNT paper, the author proved 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.
%C We have verified that a(n) > 0 for all n = 1..10^7.
%H Zhi-Wei Sun, <a href="/A300441/b300441.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e a(1) = 1 since 1^2 - 4^0*(4*u(0)^2+1) = 1 is 1^2 + 0^2.
%e a(5) = 3 since 5^2 - 4^0*(4*u(1)^2+1) = 20 = 4^2 + 2^2, 5^2 - 4^1*(4*u(1)^2+1) = 5 = 2^2 + 1^2, and 5^2 - 4^2*(4*u(0)^2+1) = 9 = 3^2 + 0^2.
%e a(7) = 1 since 7^2 - 4^1*(4*u(0)^2+1) = 45 = 6^2 + 3^2.
%t u[0]=0;
%t u[1]=1;
%t u[n_]:=u[n]=4u[n-1]-u[n-2];
%t f[n_]:=f[n]=FactorInteger[n];
%t g[n_]:=g[n]=Sum[If[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1,1,0],{i,1,Length[f[n]]}]==0;
%t QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
%t tab={};Do[r=0;Do[m=0;Label[cc];If[4u[m]^2+1>n^2/4^k,Goto[bb]];If[QQ[n^2-4^k*(4u[m]^2+1)],r=r+1,m=m+1;Goto[cc]];
%t Label[bb],{k,0,Log[2,n]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y Cf. A000118, A000290, A000302, A001353, A001481, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Mar 05 2018
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