OFFSET
1,1
COMMENTS
It is easy to find squares that are triangular, pentagonal, hexagonal, etc. So it is somewhat surprising that there are no square 10-gonal numbers other than 0 and 1. For these n, the equation 2*x^2 = (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 2k^2+2 (A005893), 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
Muniru A Asiru, Table of n, a(n) for n = 1..244
Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781-802.
MATHEMATICA
P[n_, k_]:=1/2n(n(k-2)+4-k); data1=2#^2+2&/@Range[2, 161]; data2=Head[Reduce[m^2==P[n, #] && 1<m && 1<n && !m==n, {m, n}, Integers]]&/@data1; data3=Flatten[Position[data2, Symbol]]; data1[[#]]&/@data3 (* Ant King, Mar 01 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 13 2011
STATUS
approved