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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. 5
 3, 36, 4, 210, 9801, 5, 6, 1225, 40755, 6, 55, 81, 4347, 121771, 7, 21, 225, 176, 11781, 297045, 8, 325, 9, 651, 325, 26884, 631125, 9, 10, 0, 12376, 1540, 540, 54405, 1212751, 10, 0, 196, 715, 0, 3186, 833, 100725, 2158695, 11, 105, 64, 12, 561, 18361, 5985, 1216, 174097, 3617601, 12 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS 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. Sequences A189217 and A189218 give the index of T(n,k) as a k-gonal and n-gonal number, respectively. LINKS Table of n, a(n) for n=3..57. Eric W. Weisstein, MathWorld: Polygonal Number EXAMPLE The triangle begins: 3 36, 4 210, 9801, 5 6, 1225, 40755, 6 55, 81, 4347, 121771, 7 21, 225, 176, 11781, 297045, 8 325, 9, 651, 325, 26884, 631125, 9 10, 0, 12376, 1540, 540, 54405, 1212751, 10 0, 196, 715, 0, 3186, 833, 100725, 2158695, 11 MATHEMATICA 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}]] CROSSREFS Cf. A188891, A100252. Sequence in context: A231831 A119526 A112404 * A291213 A158301 A105758 Adjacent sequences: A189213 A189214 A189215 * A189217 A189218 A189219 KEYWORD nonn,tabl AUTHOR T. D. Noe, Apr 18 2011 STATUS approved

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Last modified September 28 13:02 EDT 2023. Contains 365735 sequences. (Running on oeis4.)