A188894 -id:A188894 - cp's OEIS frontend

A188891 Least triangular n-gonal number greater than 1, or 0 if none exists.

3, 36, 210, 6, 55, 21, 325, 10, 0, 105, 36, 1275, 15, 45, 231, 0, 946, 276, 21, 11935, 66, 136, 351, 1596, 78, 28, 1225, 595, 820, 58653, 190, 325, 1335795, 36, 6670, 0, 561, 4005, 120, 1128, 1485, 203841, 45, 666, 6903, 465, 4950, 20910, 741, 153, 10731, 8911, 55, 1953

Offset: 3

T. D. Noe, Apr 13 2011

nonn

See A188893 and A188894 for the corresponding indices of these terms. Note that a(n) is zero for n = 11, 18, 38 (numbers in A188892). Although the Mathematica program searches only the first 20000 triangular numbers for n-gonal numbers, the Reduce function can show that there are no triangular n-gonal numbers (other than 0 and 1) for these n.

Eric W. Weisstein, MathWorld: Polygonal Number
Eric W. Weisstein, MathWorld: Triangular Number

Cf. A000217 (triangular numbers), A100252 (similar sequence for squares).

NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[tr = k*(k+1)/2; i = NgonIndex[n, tr]; k < 20000 && ! IntegerQ[i], k++]; If[k==20000, tr=0]; tr, {n, 3, 50}]
Table[SelectFirst[PolygonalNumber[n,Range[2,1000]],OddQ[Sqrt[8#+1]]&],{n,3,100}]/.Missing["NotFound"]->0 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 10 2019 *)

A189218 Triangle read by rows of the index of the least n-gonal number A189216(n,k).

Original entry on oeis.org

2, 6, 2, 12, 81, 2, 2, 25, 143, 2, 5, 6, 42, 221, 2, 3, 9, 8, 63, 315, 2, 10, 2, 14, 10, 88, 425, 2, 2, 0, 56, 20, 12, 117, 551, 2, 0, 7, 13, 0, 27, 14, 150, 693, 2, 5, 4, 2, 11, 61, 35, 16, 187, 851, 2, 3, 3, 4, 28, 11, 43, 44, 18, 228, 1025, 2, 15, 9, 0, 99, 1989, 609, 117, 54, 20, 273, 1215, 2

Offset: 3

T. D. Noe, Apr 18 2011

The n-th term of the n-th row is 2. The first and second columns are A188894 and A188898.

			The triangle begins:
2
6,   2
12,  81,   2
2,   25,   143,  2
5,   6,    42,   221,  2
3,   9,    8,    63,   315,  2
10,  2,    14,   10,   88,   425,  2
2,   0,    56,   20,   12,   117,  551,  2
0,   7,    13,   0,    27,   14,   150,  693,  2

cp's OEIS Frontend

A188891 Least triangular n-gonal number greater than 1, or 0 if none exists.

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A189218 Triangle read by rows of the index of the least n-gonal number A189216(n,k).

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