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 A179611 #15 Apr 07 2024 08:49:27
%S A179611 1,4,16,68,280,1168,4848,20160,83776,348224,1447296,6015488,25002240,
%T A179611 103917568,431915008,1795179520,7461349376,31011794944,128895102976,
%U A179611 535729963008,2226667929600,9254755975168,38465775239168
%N A179611 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3).
%C A179611 The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
%C A179611 The sequence above corresponds to 36 red king vectors, i.e., A[5] vectors, with decimal values 15, 39, 45, 75, 78, 99, 102, 105, 108, 135, 141, 165, 195, 198, 201, 204, 225, 228, 267, 270, 291, 294, 297, 300, 330, 354, 360, 387, 390, 393, 396, 417, 420, 450, 456 and 480.
%H A179611 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,8,4).
%F A179611 G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3).
%F A179611 a(n) = 2*a(n-1) + 8*a(n-2) + 4*a(n-3) with a(1)=1, a(2)=4 and a(3)=16.
%F A179611 a(n) = (8 + 3*z1 - 6*z1^2)*z1^(-n)/(z1*37) + (8 + 3*z2 - 6*z2^2)*z2^(-n)/(z2*37) + (8 + 3*z3 - 6*z3^2)*z3^(-n)/(z3*37) with z1, z2 and z3 the roots of f(x) = 1 - 2*x - 8*x^2 - 4*x^3 = 0.
%F A179611 alpha = arctan(3*sqrt(111));
%F A179611 z1 = sqrt(10)*cos(alpha/3)/6 + sqrt(30)*sin(alpha/3)/6 - 2/3 = 0.2405971520460078;
%F A179611 z2 = -sqrt(10)*cos(alpha/3)/3 - 2/3 = -1.585043243313016;
%F A179611 z3 = sqrt(10)*cos(alpha/3)/6 - sqrt(30)*sin(alpha/3)/6 - 2/3 = -0.6555539087329909.
%p A179611 with(LinearAlgebra): nmax:=22; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [0,0,0,0,0,1,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): 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 A179611 LinearRecurrence[{2,8,4},{1,4,16},30] (* _Harvey P. Dale_, Oct 20 2017 *)
%Y A179611 Cf. A179596, A179597 (central square).
%Y A179611 Cf. A052904.
%K A179611 easy,nonn
%O A179611 0,2
%A A179611 _Johannes W. Meijer_, Jul 28 2010