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%I #13 May 30 2018 12:19:08
%S 0,1,2,3,2,3,4,4,2,3,5,5,2,3,5,5,4,3,6,8,4,3,6,6,3,3,5,7,6,3,4,8,5,2,
%T 6,7,3,4,5,5,6,4,5,10,6,4,7,8,4,2,7,9,9,5,7,11,8,2,5,11,5,4,4,8,8,4,6,
%U 11,10,3,6,8,5,5,6,7,6,6,5,9
%N Number of ways to write n as a^2 + b^2 + C(k) + C(m) with 0 <= a <= b and 0 < k <= m, where C(k) denotes the Catalan number binomial(2k,k)/(k+1).
%C Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two squares and two Catalan numbers.
%C This is similar to the author's conjecture in A303540. It has been verified that a(n) > 0 for all n = 2..10^9.
%H Zhi-Wei Sun, <a href="/A303543/b303543.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(2) = 1 with 2 = 0^2 + 0^2 + C(1) + C(1).
%e a(3) = 2 with 3 = 0^2 + 1^2 + C(1) + C(1) = 0^2 + 0^2 + C(1) + C(2).
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t c[n_]:=c[n]=Binomial[2n,n]/(n+1);
%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=1;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,1,k}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y Cf. A000108, A000290, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303601.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Apr 25 2018
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