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
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 13 2011
STATUS
approved