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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3). 29
1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090 (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 a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.

In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.

On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.

The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the central square to A175655.

REFERENCES

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

David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.

Johannes W. Meijer, The elephant sequences.

Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565

Wikipedia, War Elephant.

Index entries for linear recurrences with constant coefficients, signature (3, 1, -6).

FORMULA

G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).

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

a(n) = ((6+10*A)*A^(-n-1) + (6+10*B)*B^(-n-1))/13-2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.

Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).

b(n) = sum(sum(binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j),j,0,k),k,1,n); n>0, b(0)=1. a(n)=b(n)-b(n-1)-b(n-2), a(0)=b(0), a(1)=b(1)-b(0). - Vladimir Kruchinin, Aug 20 2010

MAPLE

nmax:=28; m:=1; A[1]:=[0, 0, 0, 0, 1, 0, 0, 0, 1]: A[2]:=[0, 0, 0, 1, 0, 1, 0, 0, 0]: A[3]:=[0, 0, 0, 0, 1, 0, 1, 0, 0]: A[4]:=[0, 1, 0, 0, 0, 0, 0, 1, 0]: A[5]:=[0, 0, 1, 0, 0, 0, 1, 1, 1]: A[6]:=[0, 1, 0, 0, 0, 0, 0, 1, 0]: A[7]:=[0, 0, 1, 0, 1, 0, 0, 0, 0]: A[8]:=[0, 0, 0, 1, 0, 1, 0, 0, 0]: A[9]:=[1, 0, 0, 0, 1, 0, 0, 0, 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[{3, 1, -6}, {1, 2, 6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

PROG

(PARI) a(n)=([0, 1, 0; 0, 0, 1; -6, 1, 3]^n*[1; 2; 6])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

CROSSREFS

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Sequence in context: A323027 A110152 A245560 * A017922 A077937 A077981

Adjacent sequences:  A175651 A175652 A175653 * A175655 A175656 A175657

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

STATUS

approved

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Last modified March 21 07:23 EDT 2019. Contains 321367 sequences. (Running on oeis4.)