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.

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

Offset: 3

Author

T. D. Noe, Apr 18 2011

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• Eric W. Weisstein, MathWorld: Polygonal Number

Crossrefs

Cf. A188891, A100252.

Programs

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}]]