A212614 Least k > 1 such that the product tri(n) * tri(k) is triangular, or zero if no such k exists, where tri(k) is the k-th triangular number.

2, 5, 3, 6, 2, 4, 10, 0, 13, 7, 5, 4, 9, 3, 20, 208, 185, 14, 5, 2, 6, 14, 12, 115, 55, 37, 748, 11, 12, 1358, 90, 90, 6, 3, 21, 11, 26, 10, 33, 21, 265, 51, 61, 75, 96, 131, 201, 411, 0, 10, 7, 148, 113, 92, 4, 68, 364, 329, 50, 5083, 43, 329594, 38, 36, 2414

Offset

1,1

Comments

That is, tri(k) = k(k+1)/2. It is provable that a(8) and a(49) are zero.

Other terms that are zero are given in sequence A001108. Note that a(71) = 2076978. In general, a Pell equation of the form x^2 = 1 + 2*n(n+1)*y*(y+1) must be solved to find a(n). - T. D. Noe, Jun 03 2012

Links

Table of n, a(n) for n=1..65.

Example

For n = 2, tri(n) = 3 and the first k is 5 because tri(5) = 15 and 3*15 = 45 is triangular.

Mathematica

kMax = 10^6; TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; Table[t = n*(n+1)/2; k = 2; While[t2 = k*(k+1)/2; k < kMax && ! TriangularQ[t*t2], k++]; If[k == kMax, 0, k], {n, 65}]

Crossrefs

Cf. A188630 (triangular numbers that are tri(x) * tri(y) for some x,y > 1).

Cf. A212615 (similar sequence for pentagonal numbers).

Cf. A000217 (triangular numbers).

Sequence in context: A198140 A339259 A340066 * A037852 A367433 A226214

Adjacent sequences: A212611 A212612 A212613 * A212615 A212616 A212617

Keyword

nonn

Author

T. D. Noe, May 31 2012

Status

approved