A188892 Numbers n such that there is no triangular n-gonal number greater than 1.

11, 18, 38, 102, 198, 326, 486, 678, 902, 1158, 1446, 1766, 2118, 2918, 3366, 3846, 4358, 4902, 5478, 6086, 6726, 7398, 8102, 8838, 9606, 10406, 11238, 12102, 12998, 13926, 14886, 15878, 16902, 17958, 19046, 20166, 21318, 22502, 24966, 26246

Offset

1,1

Comments

It is easy to find triangular numbers that are square, pentagonal, hexagonal, etc. So it is somewhat surprising that there are no triangular 11-gonal numbers other than 0 and 1. For these n, the equation x^2 + x = (n-2)*y^2 - (n-4)*y has no integer solutions x>1 and y>1.

Chu shows how to transform the equation into a generalized Pell equation. When n has the form k^2+2 (A059100), then the Pell equation has only a finite number of solutions and it is simple to select the n that produce no integer solutions greater than 1.

The general case is in A188950.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Wenchang Chu, Regular polygonal numbers and generalized pell equations, Int. Math. Forum 2 (2007), 781-802.

Maple

filter:= n -> nops(select(t -> min(subs(t, [x, y]))>=2, [isolve(x^2 + x = (n-2)*y^2 - (n-4)*y)])) = 0:
select(filter, [seq(t^2+2, t=3..200)]); # Robert Israel, May 13 2018

Crossrefs

Cf. A051682 (11-gonal numbers), A051870 (18-gonal numbers), A188891, A188896.

Sequence in context: A151748 A003334 A037006 * A168433 A302455 A303237

Adjacent sequences: A188889 A188890 A188891 * A188893 A188894 A188895

Keyword

nonn

Author

T. D. Noe, Apr 13 2011

Status

approved