

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 kth triangular number.


2



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 A226214 A331212
Adjacent sequences: A212611 A212612 A212613 * A212615 A212616 A212617


KEYWORD

nonn


AUTHOR

T. D. Noe, May 31 2012


STATUS

approved



