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%I #26 Apr 27 2018 12:05:44
%S 0,1,2,3,3,4,4,5,5,6,4,5,7,5,4,7,7,7,8,8,5,9,10,7,6,9,8,8,6,7,10,10,9,
%T 8,7,8,9,10,6,9,11,7,6,8,9,10,7,10,8,7,8,10,10,9,10,8,9,13,14,10,11,
%U 12,12,9,9,12,11,13,11,9
%N Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + Bell(k) + Bell(m) with 0 <= a <= b and 0 < k <= m, where Bell(k) denotes the k-th Bell number A000110(k).
%C Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be expressed as the sum of two triangular numbers and two Bell numbers.
%C This has been verified for all n = 2..7*10^8. Note that 111277 cannot be written as the sum of two squares and two Bell numbers.
%C As log(Bell(n)) is asymptotically equivalent to n*log(n), Bell numbers eventually grow faster than any exponential function.
%C See also A303389, A303540, A303543 and A303637 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A303601/b303601.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://math.scichina.com:8081/sciAe/EN/abstract/abstract517007.shtml">On universal sums of polygonal numbers</a>, Sci. China Math. 58(2015), no. 7, 1367-1396.
%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.
%e a(2) = 1 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + Bell(1) + Bell(1).
%e a(3) = 2 with 3 = 0*(0+1)/2 + 1*(1+1)/2 + Bell(1) + Bell(1) = 0*(0+1)/2 + 0*(0+1)/2 + Bell(1) + Bell(2).
%t TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
%t b[n_]:=b[n]=BellB[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=1;Label[bb];If[b[k]>n,Goto[aa]];Do[If[QQ[4(n-b[k]-b[j])+1],Do[If[TQ[n-b[k]-b[j]-x(x+1)/2],r=r+1],{x,0,(Sqrt[4(n-b[k]-b[j])+1]-1)/2}]],{j,1,k}];k=k+1;Goto[bb];Label[aa];
%t tab=Append[tab,r],{n,1,70}];Print[tab]
%Y Cf. A000110, A000217, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303637.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Apr 26 2018
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