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 A180037 #33 Sep 08 2022 08:45:54
%S A180037 1,6,32,172,924,4964,26668,143268,769676,4134916,22213932,119339492,
%T A180037 641125324,3444305604,18503778668,99407504548,534045080076,
%U A180037 2869040409476,15413292207532,82804541856612,444849293698124
%N A180037 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5*x-2*x^2).
%C A180037 The a(n) represent the number of n-move routes of a fairy chess piece starting in the corner and side squares (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032.
%C A180037 The sequence above corresponds to 28 red queen vectors, i.e. A[5] vector, with decimal values between 3 and 384. The central squares lead for these vectors to A180038.
%C A180037 For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,4,5} containing no subwords 00, 11, 22 and 33. - _Milan Janjic_, Jan 31 2015, Oct 05 2016
%C A180037 a(n) equals the number of sequences over {0,1,2,3,4,5} of length n where no two consecutive terms differ by 4. - _David Nacin_, May 31 2017
%H A180037 Vincenzo Librandi, <a href="/A180037/b180037.txt">Table of n, a(n) for n = 0..200</a>
%H A180037 M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, J. Int. Seq. 18 (2015) # 15.4.7.
%H A180037 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5, 2).
%F A180037 G.f.: (1+x)/(1-5*x-2*x^2).
%F A180037 a(n) = 5*a(n-1) + 2*a(n-2) with a(0) = 1 and a(1) = 6.
%F A180037 a(n) = ((7-A)*A^(-n-1)+(7-B)*B^(-n-1))/33 with A = (-5+sqrt(33))/4 and B = (-5-sqrt(33))/4.
%p A180037 with(LinearAlgebra): nmax:=21; m:=1; A[5]:= [0,0,0,0,0,0,0,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);
%t A180037 LinearRecurrence[{5,2},{1,6},50] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o A180037 (Magma) I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011
%K A180037 easy,nonn
%O A180037 0,2
%A A180037 _Johannes W. Meijer_, Aug 09 2010