%I #7 Mar 30 2012 17:37:54
%S 1,2,5,9,27,45,63,105,135,225,405,630,315,531441,3645,1485,945,4851,
%T 1575,13041,2835,18225,295245,4095,3465,50625,2657205,11025,25515,
%U 52650,14175,17955,10395,1476225,215233605,97020,17325,150094635296999121
%N The least number k such that there are n different representations of k as the difference of two positive triangular numbers.
%C The first occurrence of n in A136107
%e a(0)=1 because there are no two positive triangular numbers whose difference is 1,
%e a(1)=2 because 3-1 = 2,
%e a(2)=5 because 6-1 = 15-10 = 5,
%e a(3)=9 because 10-1 = 15-6 = 45-36 = 9, etc.
%t f[n_] := f[n] = Block[{c = 0, k = 1}, While[k < n, If[IntegerQ[Sqrt[8 n + 4 k (k + 1) + 1]], c++ ]; k++ ]; c]; Table[ Position[ Table[ f@i, {i, 54000}], n, 1, 1], {n, 0, 30}] // Flatten
%Y Cf. A136107.
%K nonn
%O 0,2
%A _John W. Layman_ and _Robert G. Wilson v_, Dec 12 2007
%E 6 new terms from _Donovan Johnson_, Jan 21 2009
%E a(37) from _Max Alekseyev_, May 13 2009