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A188892 Numbers n such that there is no triangular n-gonal number greater than 1. 6

%I #19 May 13 2018 20:50:35

%S 11,18,38,102,198,326,486,678,902,1158,1446,1766,2118,2918,3366,3846,

%T 4358,4902,5478,6086,6726,7398,8102,8838,9606,10406,11238,12102,12998,

%U 13926,14886,15878,16902,17958,19046,20166,21318,22502,24966,26246

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

%C 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.

%C 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.

%C The general case is in A188950.

%H Robert Israel, <a href="/A188892/b188892.txt">Table of n, a(n) for n = 1..10000</a>

%H Wenchang Chu, <a href="http://www.m-hikari.com/imf-password2007/13-16-2007/chuIMF13-16-2007.pdf">Regular polygonal numbers and generalized pell equations</a>, Int. Math. Forum 2 (2007), 781-802.

%p filter:= n -> nops(select(t -> min(subs(t,[x,y]))>=2, [isolve(x^2 + x = (n-2)*y^2 - (n-4)*y)])) = 0:

%p select(filter, [seq(t^2+2,t=3..200)]); # _Robert Israel_, May 13 2018

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

%K nonn

%O 1,1

%A _T. D. Noe_, Apr 13 2011

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