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A303540 Number of ways to write n as a^2 + b^2 + binomial(2*c,c) + binomial(2*d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

%I A303540 #40 Jul 30 2022 12:08:21
%S A303540 0,1,2,3,2,2,3,4,3,2,3,6,4,2,2,4,4,2,2,5,5,5,4,4,4,4,5,6,5,5,4,5,4,4,
%T A303540 3,4,5,5,6,5,5,5,4,7,3,4,5,6,4,2,4,6,7,4,4,5,7,6,2,5,4,6,3,2,5,5,5,4,
%U A303540 4,3,7,9,6,5,6,11,7,3,4,8
%N A303540 Number of ways to write n as a^2 + b^2 + binomial(2*c,c) + binomial(2*d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.
%C A303540 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 central binomial coefficients.
%C A303540 It has been verified that a(n) > 0 for all n = 2..10^10.
%C A303540 See also A303539 and A303541 for related information.
%C A303540 Jiao-Min Lin (a student at Nanjing University) has verified a(n) > 0 for all 1 < n <= 10^11. - _Zhi-Wei Sun_, Jul 30 2022
%H A303540 Zhi-Wei Sun, <a href="/A303540/b303540.txt">Table of n, a(n) for n = 1..10000</a>
%H A303540 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 A303540 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 A303540 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 A303540 a(2) = 1 since 2 = 0^2 + 0^2 + binomial(2*0,0) + binomial(2*0,0).
%e A303540 a(10) = 2 with 10 = 2^2 + 2^2 + binomial(2*0,0) + binomial(2*0,0) = 1^2 + 1^2 + binomial(2*1,1) + binomial(2*2,2).
%e A303540 a(2435) = 1 with 2435 = 32^2 + 33^2 + binomial(2*4,4) + binomial(2*5,5).
%p A303540 N:= 100: # for a(1)..a(N)
%p A303540 A:= Vector(N):
%p A303540 for b from 0 to floor(sqrt(N)) do
%p A303540   for a from 0 to min(b, floor(sqrt(N-b^2))) do
%p A303540     t:= a^2+b^2;
%p A303540     for d from 0 do
%p A303540       s:= t + binomial(2*d,d);
%p A303540       if s > N then break fi;
%p A303540       for c from 0 to d do
%p A303540         u:= s + binomial(2*c,c);
%p A303540         if u > N then break fi;
%p A303540         A[u]:= A[u]+1;
%p A303540 od od od od:
%p A303540 convert(A,list); # _Robert Israel_, May 30 2018
%t A303540 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A303540 c[n_]:=c[n]=Binomial[2n,n];
%t A303540 f[n_]:=f[n]=FactorInteger[n];
%t A303540 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 A303540 QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
%t A303540 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 A303540 Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303541, A303543, A303601, A303639.
%K A303540 nonn
%O A303540 1,3
%A A303540 _Zhi-Wei Sun_, Apr 25 2018