%I A072221
%S A072221 1,4,25,148,865,5044,29401,171364,998785,5821348,33929305,197754484,
%T A072221 1152597601,6717831124,39154389145,228208503748,1330096633345,
%U A072221 7752371296324,45184131144601,263352415571284,1534930362283105
%N A072221 a(n) = 6*a(n-1) - a(n-2) + 2, with a(0)=1,a(1)=4.
%C A072221 The product of three consecutive triangular numbers with middle term 
               A000217(m) where m is in this sequence is a square.
%C A072221 n is in this sequence just in case the triangle with sides 3,n,n+1 has 
               integer area. Equivalently, n such that 2*(n+2)*(n-1) is a square. 
               [From James Buddenhagen (jbuddenh(AT)gmail.com), Oct 19 2008]
%C A072221 Triangular numbers that are equal to a square plus one have this sequence 
               as indices. For example, 25th triangular number is 25*26/2 = 325 
               = 18^2 + 1. [From Tanya Khovanova & Alexey Radul (tanyakh(AT)yahoo.com), 
               Aug 08 2009]
%F A072221 a(n)={3*A001541(n)-1}/2.
%F A072221 a(n)=3*A001108(n)+1. - David Scheers, Dec 25 2006
%F A072221 a(n)=-1/2+(3/4)*((3+sqrt(8))^n+(3-sqrt(8))^n) for n>=0. a(n)=floor((3/
               4)*(3+sqrt(8))^n) for n>0. - Franz Vrabec (franz.vrabec(AT)aon.at), 
               Aug 21 2006
%F A072221 G.f.: (1-3x+4x^2)/((1-x)(1-6x+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Sep 09 2008]
%t A072221 a[n_] := a[n] = 6a[n - 1] - a[n - 2] + 2; a[0] = 1; a[1] = 4; Table[ 
               a[n], {n, 0, 20}]
%Y A072221 Cf. A000217, A001108, A001541.
%Y A072221 Sequence in context: A156701 A015533 A079291 this_sequence A055846 A091634 
               A010909
%Y A072221 Adjacent sequences: A072218 A072219 A072220 this_sequence A072222 A072223 
               A072224
%K A072221 nonn
%O A072221 0,2
%A A072221 Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 04 2002
%E A072221 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 08 2002