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%I #40 Feb 13 2023 09:02:47
%S 1,2,6,14,36,86,210,500,1194,2822,6660,15638,36642,85604,199626,
%T 464630,1079892,2506550,5811762,13462484,31159914,72071654,166599972,
%U 384912086,888906306,2052031172,4735527306,10925175254,25198866036,58108609526,133973643090
%N 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).
%C a(n) represents 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.
%C 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.
%C 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.
%C 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.
%D Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
%D David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.
%H Vincenzo Librandi, <a href="/A175654/b175654.txt">Table of n, a(n) for n = 0..1000</a>
%H Viswanathan Anand, <a href="http://www.time.com/time/specials/2007/article/0,28804,1815747_1815707_1815674,00.html">The Indian Defense</a>, Time, Jun 19 2008.
%H Johannes W. Meijer, <a href="/A175654/a175654.jpg">The elephant sequences</a>.
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/War_elephant">War Elephant</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-6).
%F G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).
%F a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=2 and a(2)=6.
%F 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.
%F Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).
%F a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} (Sum_{j=0..k} (binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - _Vladimir Kruchinin_, Aug 20 2010
%F a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - _G. C. Greubel_, Dec 08 2021
%F E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - _Stefano Spezia_, Feb 12 2023
%p 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);
%t LinearRecurrence[{3,1,-6}, {1,2,6}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 21 2012 *)
%o (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
%o (Magma) [n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // _G. C. Greubel_, Dec 08 2021
%o (Sage) [( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # _G. C. Greubel_, Dec 08 2021
%Y 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].
%Y Cf. A000244, A006130, A006138, A075188, A175655, A180140.
%K easy,nonn
%O 0,2
%A _Johannes W. Meijer_, Aug 06 2010; edited Jun 21 2013
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