This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181394 #23 Jan 03 2022 07:36:45
%S A181394 2,14,64,284,1206,4882,19060,72588,271548,1001964,3656480,13223348,
%T A181394 47461350,169263658,600355808,2119297852,7450253362,26095036854,
%U A181394 91102304600,317127751352,1101029901244,3813576283628,13180379580636,45463936339816
%N A181394 Summed lengths of all nonintersecting rook paths on a 3 x n board.
%C A181394 Paths are self-avoiding from one corner to the diagonally opposite corner.
%H A181394 Alois P. Heinz, <a href="/A181394/b181394.txt">Table of n, a(n) for n = 1..1000</a>
%H A181394 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-22,28,-23,4,2,-4,-1).
%F A181394 G.f.: 2*x*(1 - x - 2*x^2 + 12*x^3 - 2*x^4 + 2*x^5 - 2*x^6)/((1 - x + x^2)^2*(1 - 3*x - x^2)^2). - _Alois P. Heinz_, Nov 26 2010, modified _Andrew Howroyd_, Jan 06 2020
%F A181394 Asymptotics: a(n) ~ ((3/4-sqrt(13)/52)*n-1/4-sqrt(13)/52)*((sqrt(13)+3)/2)^n. - _Vaclav Kotesovec_, Aug 31 2012
%e A181394 E.g. s(2) {RRD, DRR, RDR, DRURD} 3+3+3+5 = 14.
%p A181394 a:= n-> (Matrix(8, (i,j)->if i+1=j then 1 elif i=8 then [-1, -4, 2, 4, -23, 28, -22, 8][j] else 0 fi)^n. <<0, 2, 14, 64, 284, 1206, 4882, 19060>>)[1, 1]: seq (a(n), n=1..24); # _Alois P. Heinz_, Nov 26 2010
%t A181394 LinearRecurrence[{8, -22, 28, -23, 4, 2, -4, -1}, {2, 14, 64, 284, 1206, 4882, 19060, 72588}, 24] (* _Jean-François Alcover_, Jan 03 2022 *)
%Y A181394 Enumeration of these paths is A006192, related sequences A181395, A181396, A181397, A181398, A181399.
%K A181394 nonn,walk
%O A181394 1,1
%A A181394 _David Scambler_, Oct 17 2010
%E A181394 More terms from _Alois P. Heinz_, Nov 26 2010