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%I #8 Feb 28 2019 11:44:47
%S 1,2,1,1,4,4,1,2,5,6,4,2,5,9,4,1,6,7,4,4,4,5,4,1,4,8,4,1,6,7,3,0,3,6,
%T 7,3,3,9,5,3,7,5,3,6,7,6,5,1,7,11,4,2,7,9,5,2,3,7,9,4,3,10,6,1,8,7,3,
%U 6,6,6,6,2,4,11,4,3,5,5,4,1,4,5,9,3,6,12,5,0,7,10,6,5,3,10,9,0,2,11,5,4
%N Number of ways to write n as w^2 + (x*(x+1)/2)^2 + (y*(3y+1)/2)^2 + (z*(5z+1)/2)^2, where w,x,y,z are integers with w > 0 and x >= 0.
%C Conjecture 1 (Big 1-3-5 Conjecture): a(n) > 0 for any positive integer n not divisible by 8.
%C This is stronger than Lagrange's four-square theorem, but different from the 1-3-5 conjecture in A271518. I'd like to call it "the Big 1-3-5 Conjecture". I have verified a(n) > 0 for any positive integer n < 3*10^6 not divisible by 8.
%C Conjecture 2: If c is 5 or 7, then any positive integer n not divisible by 8 can be written as w^2 + (x*(x+1)/2)^2 + (y(3*y+1)/2)^2 + (z*(c*z+3)/2)^2, where w,x,y,z are integers with w > 0.
%C Conjecture 3: If a positive integer n is neither 76 nor divisible by 8, then it can be written as w^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(5z+3)/2)^2, where w,x,y,z are integers with w > 0.
%C Conjecture 4: Each n = 0,1,2,... can be written as 2*w^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(5z+3)/2)^2 with w,x,y,z integers.
%H Zhi-Wei Sun, <a href="/A306614/b306614.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.
%e a(3) = 1 with 3 = 1^2 + (1*(1+1)/2)^2 + ((-1)*(3*(-1)+1)/2)^2 + (0*(5*0+1)/2)^2.
%e a(4) = 1 with 4 = 2^2 + (0*(0+1)/2)^2 + (0*(3*0+1)/2)^2 + (0*(5*0+1)/2)^2.
%e a(7) = 1 with 7 = 1^2 + (1*(1+1)/2)^2 + ((-1)*(3*(-1)+1)/2)^2 + ((-1)*(5*(-1)+1)/2)^2.
%e a(28) = 1 with 28 = 3^2 + (2*(2+1)/2)^2 + ((-1)*(3*(-1)+1)/2)^2 + (1*(5*1+1)/2)^2.
%t t[x_]:=t[x]=x(x+1)/2; p[x_]:=p[x]=x(3x+1)/2; q[x_]:=q[x]=x(5x+1)/2; SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[n-t[x]^2-p[y]^2-q[z]^2],r=r+1],{x,0,(Sqrt[8*Sqrt[n-1]+1]-1)/2},{y,-Floor[(Sqrt[24*Sqrt[n-1-t[x]^2]+1]+1)/6],(Sqrt[24*Sqrt[n-1-t[x]^2]+1]-1)/6}, {z,-Floor[(Sqrt[40*Sqrt[n-1-t[x]^2-p[y]^2]+1]+1)/10],(Sqrt[40*Sqrt[n-1-t[x]^2-p[y]^2]+1]-1)/10}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y Cf. A000118, A000217, A000290, A001318, A057569, A271518.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Feb 28 2019
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