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 A179606 #22 Aug 01 2022 08:08:51
%S A179606 1,4,17,71,298,1249,5237,21956,92053,385939,1618082,6783941,28442233,
%T A179606 119246404,499950377,2096083151,8788001338,36844419769,154473265997,
%U A179606 647641896836,2715292020493,11384085545659,47728716739442
%N A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3*x - 5*x^2).
%C A179606 a(n) represents 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 A179606 The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the side squares to A152187.
%C A179606 This sequence belongs to a family of sequences with g.f. (1 + (k-4)*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k= 2), A015521 (k=4), A179606 (k=5; this sequence), A154964 (k=6), A179603 (k=7) and A179599 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A006190 (k=1), A133494 (k=0) and A168616 (k=-2).
%C A179606 Inverse binomial transform of A052918.
%C A179606 The sequence b(n+1) = 6*a(n), n >= 0 with b(0)=1, is a berserker sequence, see A180147. The b(n) sequence corresponds to 16 A[5] vectors with decimal values between 111 and 492. These vectors lead for the corner squares to sequence c(n+1)=4*A179606(n), n >= 0 with c(0)=1, and for the side squares to A180140. - _Johannes W. Meijer_, Aug 14 2010
%C A179606 Equals the INVERT transform of A063782: (1, 3, 10, 32, 104, ...). Example: a(3) = 71 = (1, 1, 4, 7) dot (32, 10, 3, 1) = (32 + 10 + 12 + 17). - _Gary W. Adamson_, Aug 14 2010
%H A179606 Indranil Ghosh, <a href="/A179606/b179606.txt">Table of n, a(n) for n = 0..1603</a>
%H A179606 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,5).
%F A179606 G.f.: (1+x)/(1 - 3*x - 5*x^2).
%F A179606 a(n) = A015523(n) + A015523(n+1).
%F A179606 a(n) = 3*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 4.
%F A179606 a(n) = ((29 + 7*sqrt(29))*A^(-n-1) + (29-7*sqrt(29))*B^(-n-1))/290 with A = (-3+sqrt(29))/10 and B = (-3-sqrt(29))/10
%F A179606 Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A000351(n)*A130196(n)/(A015523(n)*sqrt(29) - A072263(n)) for n >= 1.
%p A179606 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]:= [0,0,0,1,1,1,0,0,1]: 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 A179606 CoefficientList[Series[(1+x)/(1-3*x-5*x^2), {x, 0, 22}],x] (* or *) LinearRecurrence[{3,5,0},{1,4},23] (* _Indranil Ghosh_, Mar 05 2017 *)
%o A179606 (PARI) print(Vec((1 + x)/(1- 3*x - 5*x^2) + O(x^23))); \\ _Indranil Ghosh_, Mar 05 2017
%Y A179606 Cf. A179597 (central square).
%Y A179606 Cf. A072263, A072264, A152187, A197189.
%K A179606 nonn,easy
%O A179606 0,2
%A A179606 _Johannes W. Meijer_, Jul 28 2010