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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).

%I A175654 #48 Feb 08 2025 09:01:30
%S A175654 1,2,6,14,36,86,210,500,1194,2822,6660,15638,36642,85604,199626,
%T A175654 464630,1079892,2506550,5811762,13462484,31159914,72071654,166599972,
%U A175654 384912086,888906306,2052031172,4735527306,10925175254,25198866036,58108609526,133973643090
%N 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).
%C A175654 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 A175654 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 A175654 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 A175654 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 A175654 Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
%D A175654 David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.
%H A175654 Vincenzo Librandi, <a href="/A175654/b175654.txt">Table of n, a(n) for n = 0..1000</a>
%H A175654 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 A175654 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H A175654 Johannes W. Meijer, <a href="/A175654/a175654.jpg">The elephant sequences</a>.
%H A175654 Wikipedia, <a href="http://en.wikipedia.org/wiki/War_elephant">War Elephant</a>.
%H A175654 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-6).
%F A175654 G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).
%F A175654 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 A175654 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 A175654 Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).
%F A175654 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 A175654 a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - _G. C. Greubel_, Dec 08 2021
%F A175654 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 A175654 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 A175654 LinearRecurrence[{3,1,-6}, {1,2,6}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 21 2012 *)
%o A175654 (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 A175654 (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 A175654 (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 A175654 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 A175654 Cf. A000244, A006130, A006138, A075188, A175655, A180140.
%K A175654 easy,nonn
%O A175654 0,2
%A A175654 _Johannes W. Meijer_, Aug 06 2010; edited Jun 21 2013