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A303235 Number of ordered pairs (x, y) with 0 <= x <= y such that n - 2^x - 2^y can be written as the sum of two triangular numbers.

%I A303235 #9 Apr 20 2018 13:18:08
%S A303235 0,1,2,3,4,5,4,5,6,6,6,8,7,7,8,8,8,10,10,10,10,9,9,11,9,10,11,10,9,12,
%T A303235 10,11,14,13,11,14,12,12,13,15,12,14,12,13,14,14,14,15,13,11,14,13,11,
%U A303235 16,13,10,11,13,11,14
%N A303235 Number of ordered pairs (x, y) with 0 <= x <= y such that n - 2^x - 2^y can be written as the sum of two triangular numbers.
%C A303235 Conjecture: a(n) > 0 for all n > 1.
%C A303235 Note that a nonnegative integer m is the sum of two triangular numbers if and only if 4*m+1 (or 8*m+2) can be written as the sum of two squares.
%C A303235 We have verified a(n) > 0 for all n = 2..4*10^8. See also the related sequences A303233 and A303234.
%H A303235 Zhi-Wei Sun, <a href="/A303235/b303235.txt">Table of n, a(n) for n = 1..10000</a>
%H A303235 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 A303235 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 A303235 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 A303235 a(2) = 1 with 2 - 2^0 - 2^0 = 0*(0+1)/2 + 0*(0+1)/2.
%e A303235 a(3) = 2 with 3 - 2^0 - 2^0 = 0*(0+1)/2 + 1*(1+1)/2 and 3 - 2^0 - 2^1 = 0*(0+1)/2 + 0*(0+1)/2.
%t A303235 f[n_]:=f[n]=FactorInteger[n];
%t A303235 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 A303235 QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
%t A303235 tab={};Do[r=0;Do[If[QQ[4(n-2^k-2^j)+1],r=r+1],{k,0,Log[2,n]-1},{j,k,Log[2,n-2^k]}];tab=Append[tab,r],{n,1,60}];Print[tab]
%Y A303235 Cf. A000079, A000217, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234.
%K A303235 nonn
%O A303235 1,3
%A A303235 _Zhi-Wei Sun_, Apr 20 2018