

A189216


Triangle T(n,k) read by rows of the smallest ngonal number greater than 1 that is also kgonal, 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
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OFFSET

3,1


COMMENTS

The first column (k=3, triangular numbers) is A188891. The second column (k=4, squares) is A100252. The nth term of the nth 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 (k2)*x^2  (k4)*x = (n2)*y^2  (n4)*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 kgonal and ngonal number, respectively.


LINKS



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



KEYWORD



AUTHOR



STATUS

approved



