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 A180035 #30 Oct 12 2022 08:41:17
%S A180035 1,6,33,183,1014,5619,31137,172542,956121,5298231,29359518,162692283,
%T A180035 901539969,4995776694,27683503377,153404846967,850074744966,
%U A180035 4710588265731,26103165563553,144647592614958,801547459765449
%N A180035 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5*x-3*x^2).
%C A180035 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 A180035 The sequence above corresponds to 56 red queen vectors, i.e. A[5] vector, with decimal values between 7 and 448. The central squares lead for these vectors to A180036.
%C A180035 For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5} containing no subwords 00, 11 and 22. - _Milan Janjic_, Jan 31 2015
%H A180035 Vincenzo Librandi, <a href="/A180035/b180035.txt">Table of n, a(n) for n = 0..200</a>
%H A180035 D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 17.
%H A180035 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5, 3).
%F A180035 G.f.: (1+x)/(1-5*x-3*x^2).
%F A180035 a(n) = 5*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 6.
%F A180035 a(n) = ((7+A)*A^(-n-1)+(7+B)*B^(-n-1))/37 with A = (-5+sqrt(37))/6 and B = (-5-sqrt(37))/6.
%F A180035 a(n) = Sum_{k, 0<=k<=n} A202396(n,k)*2^k. - _Philippe Deléham_, Dec 21 2011
%F A180035 a(n) = A015536(n+1)+A015536(n). - _R. J. Mathar_, Aug 04 2019
%p A180035 with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [0,0,0,0,0,0,1,1,1]: 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 A180035 LinearRecurrence[{5,3},{1,6},50] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o A180035 (Magma) I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011
%K A180035 easy,nonn
%O A180035 0,2
%A A180035 _Johannes W. Meijer_, Aug 09 2010