%I #34 Mar 03 2024 22:19:40
%S 0,10,36,68,108,150,200,252,312,374,444,516,596,678,768,860,960,1062,
%T 1172,1284,1404,1526,1656,1788,1928,2070,2220,2372,2532,2694,2864,
%U 3036,3216,3398,3588,3780,3980,4182,4392,4604,4824,5046,5276,5508,5748,5990,6240,6492,6752,7014,7284,7556
%N Number of ways to choose three distinct points from a 3 X n grid so that they form an isosceles triangle.
%H Chai Wah Wu, <a href="http://arxiv.org/abs/1605.00180">Counting the number of isosceles triangles in rectangular regular grids</a>, arXiv:1605.00180 [math.CO], 2016.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, -2, 1).
%F Conjectured g.f.: 2*x*(2*x^4+4*x^3+2*x^2-8*x-5)/((x+1)*(x-1)^3).
%F Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 6.
%F Conjectures from _Colin Barker_, Apr 25 2016: (Start)
%F a(n) = (-3*(47+(-1)^n)+64*n+10*n^2)/4 for n>2.
%F a(n) = (5*n^2+32*n-72)/2 for n>2 and even.
%F a(n) = (5*n^2+32*n-69)/2 for n>2 and odd.
%F (End)
%F The conjectured g.f. and recurrence are true. See paper in links. - _Chai Wah Wu_, May 07 2016
%e n=2: Label the points
%e 1 2 3
%e 4 5 6
%e There are 8 small isosceles triangles like 124 plus 135 and 246, so a(2) = 10.
%t Join[{0, 10}, LinearRecurrence[{2, 0, -2, 1}, {36, 68, 108, 150}, 50]] (* _Jean-François Alcover_, Oct 10 2018 *)
%Y Row 3 of A271910.
%Y Cf. A186434, A187452.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Apr 24 2016
%E More terms from _Jean-François Alcover_, Oct 10 2018