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A175654
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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).
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29
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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
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OFFSET
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0,2
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COMMENTS
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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 central 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 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 (we assume here that the raging elephant might behave like a bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the cross-references for some examples.
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.
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REFERENCES
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Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, June 19, 2008.
Wikipedia, War Elephant.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
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GF(x) = (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.
Limit(a(n+k)/a(k), k=infinity) = (-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). [From Vladimir Kruchinin, Aug 20 2010]
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MAPLE
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with(LinearAlgebra): nmax:=28; m:=1; A[5]:= [0, 0, 1, 0, 0, 0, 1, 1, 1]: A:=Matrix([[0, 0, 0, 0, 1, 0, 0, 0, 1], [0, 0, 0, 1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0], A[5], [0, 1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0]]): 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);
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MATHEMATICA
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LinearRecurrence[{3, 1, -6}, {1, 2, 6}, 80] (* From Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)
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CROSSREFS
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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: A178320 A025257 A110152 * A017922 A077937 A077981
Adjacent sequences: A175651 A175652 A175653 * A175655 A175656 A175657
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KEYWORD
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easy,nonn
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AUTHOR
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Johannes W. Meijer, Aug 06 2010, Aug 10 2010
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STATUS
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approved
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