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A188892 Numbers n such that there is no triangular n-gonal number greater than 1. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 15 05:50 EST 2019. Contains 329144 sequences. (Running on oeis4.)