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A189216 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.

%I A189216 #11 Feb 16 2025 08:33:14
%S A189216 3,36,4,210,9801,5,6,1225,40755,6,55,81,4347,121771,7,21,225,176,
%T A189216 11781,297045,8,325,9,651,325,26884,631125,9,10,0,12376,1540,540,
%U A189216 54405,1212751,10,0,196,715,0,3186,833,100725,2158695,11,105,64,12,561,18361,5985,1216,174097,3617601,12
%N A189216 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 A189216 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 A189216 Sequences A189217 and A189218 give the index of T(n,k) as a k-gonal and n-gonal number, respectively.
%H A189216 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">MathWorld: Polygonal Number</a>
%e A189216 The triangle begins:
%e A189216 3
%e A189216 36,      4
%e A189216 210,     9801,    5
%e A189216 6,       1225,    40755,   6
%e A189216 55,      81,      4347,    121771,  7
%e A189216 21,      225,     176,     11781,   297045,  8
%e A189216 325,     9,       651,     325,     26884,   631125,  9
%e A189216 10,      0,       12376,   1540,    540,     54405,   1212751, 10
%e A189216 0,       196,     715,     0,       3186,    833,     100725,  2158695,  11
%t A189216 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 A189216 Cf. A188891, A100252.
%K A189216 nonn,tabl
%O A189216 3,1
%A A189216 _T. D. Noe_, Apr 18 2011