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%I #9 Apr 30 2013 21:43:52
%S 3,11,4,10,6,11,5,14,3,18,4,20,6,18,7,22,11,18,10,20,6,27,5,29,8,26,
%T 11,27,9,30,3,38,14,29,6,38,10,34,18,27,11,38,7,47,12,42,20,34,5,50,4,
%U 52,18,38,6,51,13,46,11,51,8,56,14,50,27,38,15,54,22,47
%N Pairs of numbers (n,k) such that there is no n-gonal k-gonal number greater than 1, sorted by the sum n+k and then n.
%C These are n and k such that the generalized Pell equation (k-2)*x^2 - (k-4)*x = (n-2)*y^2 - (n-4)*y has no solution in integers x>1 and y>1. The paper by Chu shows how to solve these equations. A necessary condition for a pair to be in this sequence is (n-2)(k-2) is a square. These (n,k) pairs indicate where the zeros are in triangle A189216, which gives the least n-gonal k-gonal number greater than 1. For triangular (n=3) and square (n=4) numbers, see A188892 and A188896 for lists of k.
%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.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PolygonalNumber.html">MathWorld: Polygonal Number</a>
%e The pairs begin (3,11), (4,10), (6,11), (5,14), (3,18), (4,20), (6,18).
%t maxSum=100; Reap[Do[k=s-n; If[k>n && IntegerQ[Sqrt[(n-2)*(k-2)]] && FindInstance[(k-2)*x^2 - (k-4)*x == (n-2)*y^2 - (n-4)*y && x>1 && y>1, {x,y}, Integers] == {}, Sow[{n,k}]], {s,7,maxSum}, {n,3,s-3}]][[2,1]]
%Y Cf. A188892, A188896.
%K nonn,tabf
%O 1,1
%A _T. D. Noe_, Apr 20 2011
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