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%I #12 May 03 2018 02:46:24
%S 0,1,1,2,1,3,2,2,1,2,3,3,3,3,4,2,2,2,3,4,4,5,2,4,1,2,3,3,5,3,5,1,3,1,
%T 1,6,3,8,3,6,2,4,4,2,7,5,6,2,5,2,4,5,4,8,4,7,2,4,1,3,6,4,7,3,5,2,4,2,
%U 4,9,5,6,2,6,4,5,4,7,5,2
%N Number of ways to write n as a^2 + b^2 + binomial(2*c+1,c) + binomial(2*d+1,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.
%C Conjecture: a(n) > 0 for all n > 1.
%C This is similar to the author's conjecture in A303540.
%C It has been verified that a(n) > 0 for all n = 2..6*10^8.
%H Zhi-Wei Sun, <a href="/A303639/b303639.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://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
%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(9) = 1 with 9 = 1^2 + 2^2 + binomial(2*0+1,0) + binomial(2*1+1,1).
%e a(2530) = 1 with 2530 = 0^2 + 49^2 + binomial(2*1+1,1) + binomial(2*4+1,4).
%e a(3258) = 1 with 3258 = 22^2 + 52^2 + binomial(2*3+1,3) + binomial(2*3+1,3).
%e a(5300) = 1 with 5300 = 10^2 + 59^2 + binomial(2*1+1,1) + binomial(2*6+1,6).
%e a(13453) = 1 with 13453 = 51^2 + 104^2 + binomial(2*0+1,0) + binomial(2*3+1,3).
%e a(20964) = 1 with 20964 = 13^2 + 138^2 + binomial(2*3+1,3) + binomial(2*6+1,6).
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t c[n_]:=c[n]=Binomial[2n+1,n];
%t f[n_]:=f[n]=FactorInteger[n];
%t g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
%t QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
%t tab={};Do[r=0;k=0;Label[bb];If[c[k]>n,Goto[aa]];Do[If[QQ[n-c[k]-c[j]],Do[If[SQ[n-c[k]-c[j]-x^2],r=r+1],{x,0,Sqrt[(n-c[k]-c[j])/2]}]],{j,0,k}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y Cf. A000290, A001481, A001700, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Apr 27 2018
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