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A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+3*x)/(1-6*x-3*x^2). 9
1, 9, 57, 369, 2385, 15417, 99657, 644193, 4164129, 26917353, 173996505, 1124731089, 7270376049, 46996449561, 303789825513, 1963728301761, 12693739287105, 82053620627913, 530402941628793, 3428578511656497 (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 center square (m =5) 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.

On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the center square (off the center square the piece behaves like a normal queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the center square the 512 red queens lead to 17 red queen sequences, see the overview of red queen sequences and the crossreferences.

The sequence above corresponds to just one red queen vector, i.e. A[5] =[111 111 111] vector. The other squares lead for this vector to A090018.

This sequence belongs to a family of sequences with GF(x)= (1+k*x)/(1-6*x-k*x^2). The members of this family that are red queen sequences are A180028 (k=3; this sequence), A180029 (k=2), A015451 (k=1), A000400 (k=0), A001653 (k=-1), A180034 (k=-2), A084120 (k=-3), A154626 (k=-4) and A000012 (k=-5). Other members of this family are A123362 (k=5), 6*A030192(k=-6).

Inverse binomial transform of A107903.

REFERENCES

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Tim Burton, Alice in Wonderland, 2010.

Johannes W. Meijer, The red queen sequences.

FORMULA

GF(x) = (1+3*x)/(1-6*x-3*x^2)

a(n) = 6*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 9.

a(n) = ((1-A)*A^(-n-1)+(1-B)*B^(-n-1))/4 with A=(-1+2*sqrt(3)/3) and B=(-1-2*sqrt(3)/3).

Limit(a(n+k)/a(k), k=infinity) = (-1)^(n-1)*A108411(n+1)/(A041017(n-1)*sqrt(12) - A041016(n-1)) for n>=1.

MAPLE

nmax:=19; m:=5; A[1]:=[0, 1, 1, 1, 1, 0, 1, 0, 1]: A[2]:=[1, 0, 1, 1, 1, 1, 0, 1, 0]: A[3]:=[1, 1, 0, 0, 1, 1, 1, 0, 1]: A[4]:=[1, 1, 0, 0, 1, 1, 1, 1, 0]: A[5]:=[1, 1, 1, 1, 1, 1, 1, 1, 1]: A[6]:=[0, 1, 1, 1, 1, 0, 0, 1, 1]: A[7]:=[1, 0, 1, 1, 1, 0, 0, 1, 1]: A[8]:=[0, 1, 0, 1, 1, 1, 1, 0, 1]: A[9]:=[1, 0, 1, 0, 1, 1, 1, 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

LinearRecurrence[{6, 3}, {1, 9}, 50] (* Vincenzo Librandi, Nov 15 2011 *)

PROG

(MAGMA) I:=[1, 9]; [n le 2 select I[n] else 6*Self(n-1)+3*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 15 2011

CROSSREFS

Cf. A180140 (berserker sequences)

Cf. A180032 (Corner and side squares).

Cf. Red queen sequences center square [decimal value A[5]]: A180028 [511], A180029 [255], A180031 [495], A015451 [127], A152240 [239], A000400 [63], A057088 [47], A001653 [31], A122690 [15], A180034 [23], A180036 [7], A084120 [19], A180038 [3], A154626 [17], A015449 [1], A000012 [16], A000007 [0].

Sequence in context: A026896 A080961 A163919 * A155605 A199485 A102303

Adjacent sequences:  A180025 A180026 A180027 * A180029 A180030 A180031

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, Aug 09 2010; edited Jun 21 2013

STATUS

approved

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Last modified April 24 12:03 EDT 2014. Contains 240983 sequences.