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%I #10 Mar 30 2012 17:22:58
%S 3,36,4,210,9801,5,6,1225,40755,6,55,81,4347,121771,7,21,225,176,
%T 11781,297045,8,325,9,651,325,26884,631125,9,10,0,12376,1540,540,
%U 54405,1212751,10,0,196,715,0,3186,833,100725,2158695,11,105,64,12,561,18361,5985,1216,174097,3617601,12
%N Triangle T(n,k) read by rows of the smallest n-gonal number greater than 1 that is also k-gonal, or 0 if none exists, for 3 <= k <= n.
%C The first column (k=3, triangular numbers) is A188891. The second column (k=4, squares) is A100252. The n-th term of the n-th row is n. Observe that 0 occurs for (10,4)-gonal, (11,3)-gonal, and (11,6)-gonal numbers. This can be proved by trying to solve the equation (k-2)*x^2 - (k-4)*x = (n-2)*y^2 - (n-4)*y for integers x>1 and y>1. Other pairs that are zero: (14,5), (18,3), (18,6), (18,11), (20,4), and (20,10). See A188950 for a longer list of pairs.
%C Sequences A189217 and A189218 give the index of T(n,k) as a k-gonal and n-gonal number, respectively.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PolygonalNumber.html">MathWorld: Polygonal Number</a>
%e The triangle begins:
%e 3
%e 36, 4
%e 210, 9801, 5
%e 6, 1225, 40755, 6
%e 55, 81, 4347, 121771, 7
%e 21, 225, 176, 11781, 297045, 8
%e 325, 9, 651, 325, 26884, 631125, 9
%e 10, 0, 12376, 1540, 540, 54405, 1212751, 10
%e 0, 196, 715, 0, 3186, 833, 100725, 2158695, 11
%t nn = 12; Clear[poly]; Do[poly[n] = Table[i*((n - 2)*i - (n - 4))/2, {i, 2, 20000}], {n, 3, nn}]; Flatten[Table[If[k == n, n, int = Intersection[poly[n], poly[k]]; If[int == {}, 0, int[[1]]]], {n, 3, nn}, {k, 3, n}]]
%Y Cf. A188891, A100252.
%K nonn,tabl
%O 3,1
%A _T. D. Noe_, Apr 18 2011
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