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

Author

T. D. Noe, Apr 13 2011

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

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

Programs

Mathematica

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 *)