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%I #30 Mar 03 2024 22:17:41
%S 0,4,10,16,24,32,42,52,64,76,90,104,120,136,154,172,192,212,234,256,
%T 280,304,330,356,384,412,442,472,504,536,570,604,640,676,714,752,792,
%U 832,874,916,960,1004,1050,1096,1144,1192,1242,1292,1344,1396,1450,1504
%N Number of ways to choose three distinct points from a 2 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^2-x-2)/((x+1)*(x-1)^3). It would be nice to have a proof!
%F Conjectures from _Colin Barker_, Apr 24 2016: (Start)
%F a(n) = (-1+(-1)^n+16*n+2*n^2)/4, or equivalently, a(n) = (n^2+8*n)/2 if n even, (n^2+8*n-1)/2 if n odd.
%F a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4.
%F (End)
%F The conjectured g.f. and recurrence are true. See paper in links. - _Chai Wah Wu_, May 07 2016
%e n=3: 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(3) = 10.
%t LinearRecurrence[{2,0,-2,1},{0,4,10,16},60] (* _Harvey P. Dale_, May 10 2018 *)
%Y Row 2 of A271910.
%Y Cf. A186434, A187452.
%Y Same start as, but totally different from, 2*A213707.
%K nonn,more
%O 1,2
%A _N. J. A. Sloane_, Apr 24 2016
%E More terms from _Harvey P. Dale_, May 10 2018