%I
%S 1,2,9,7,14,37,64,68,57,119,112,168,194,147,267,259,222,477,427,404,
%T 519,652,567,497,512,749,722,719,952,1209,904,1139,1267,1184,1069,
%U 1737,1594,1667,1734,2077,1799,1659,1729,1814,1762,1862,2577,2444,2997,2072,2457,2842,3029,3249,3094,3589,3999,4208,3479,3232
%N Least positive integer m such that m can be written as x*(x+1) + y*(3*y+1)/2 + z*(3*z1)/2 in exactly n ways, where x,y,z are nonnegative integers.
%C Conjecture: (i) a(n) exists for any n > 0. Moreover, n^2 is the main term of a(n) as n tends to the infinity.
%C (ii) a(n) is never divisible by 5. Also, for any n > 1 the term a(n) is not congruent to 1 modulo 5.
%H ZhiWei Sun, <a href="/A254595/b254595.txt">Table of n, a(n) for n = 1..200</a>
%H ZhiWei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, arXiv:0905.0635 [math.NT], 20092015.
%e a(3) = 9 since 9 is the first positive integer m with A254573(m)=3. Note that 9 = 2*3 +1*(3*1+1)/2 + 1*(3*11)/2 = 1*2 + 1*(3*1+1)/2 + 2*(3*21)/2 = 1*2 + 2*(3*2+1)/2 + 0*(3*01)/2.
%t sQ[n_]:=IntegerQ[Sqrt[4n+1]]
%t Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[sQ[my(3y+1)/2z(3z1)/2],r=r+1;If[r>n,Goto[aa]]],{y,0,(Sqrt[24m+1]1)/6}, {z,0,(Sqrt[24(my(3y+1)/2)+1]+1)/6}]; If[r==n,Print[n," ",m];Goto[bb],Goto[aa]]]; Label[bb];Continue,{n,1,60}]
%Y Cf. A000326, A002378, A005449, A254573, A254617.
%K nonn
%O 1,2
%A _ZhiWei Sun_, Feb 02 2015
