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A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3*x - 5*x^2). 13
1, 4, 17, 71, 298, 1249, 5237, 21956, 92053, 385939, 1618082, 6783941, 28442233, 119246404, 499950377, 2096083151, 8788001338, 36844419769, 154473265997, 647641896836, 2715292020493, 11384085545659, 47728716739442 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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.

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).

Inverse binomial transform of A052918.

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

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

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..1603

Index entries for linear recurrences with constant coefficients, signature (3,5).

FORMULA

G.f.: (1+x)/(1 - 3*x - 5*x^2).

a(n) = A015523(n) + A015523(n+1).

a(n) = 3*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 4.

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

Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*A000351(n)*A130196(n)/(A015523(n)*sqrt(29) - A072263(n)) for n >= 1.

MAPLE

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);

MATHEMATICA

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 *)

PROG

(PARI) print(Vec((1 + x)/(1- 3*x - 5*x^2) + O(x^23))); \\ Indranil Ghosh, Mar 05 2017

CROSSREFS

Cf. A179597 (central square).

Cf. A072263, A072264, A152187, A197189.

Sequence in context: A017956 A136792 A188482 * A108929 A297578 A022031

Adjacent sequences:  A179603 A179604 A179605 * A179607 A179608 A179609

KEYWORD

nonn,easy

AUTHOR

Johannes W. Meijer, Jul 28 2010

STATUS

approved

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Last modified December 15 09:05 EST 2019. Contains 329995 sequences. (Running on oeis4.)