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A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).

%I A175655 #25 Dec 25 2024 13:25:55
%S A175655 1,4,8,22,50,124,290,694,1628,3838,8978,21004,48962,114022,265004,
%T A175655 615262,1426658,3305212,7650722,17697430,40911740,94528318,218312114,
%U A175655 503994220,1163124866,2683496134,6189647948,14273690782
%N A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).
%C A175655 a(n) represents the number of n-move routes of a fairy chess piece starting in the central square (m = 5) 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 turns into a raging elephant, see A175654.
%C A175655 For the central square the 512 elephants lead to 46 different elephant sequences, see the cross-references for examples.
%C A175655 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 corner squares to A175654.
%H A175655 Vincenzo Librandi, <a href="/A175655/b175655.txt">Table of n, a(n) for n = 0..1000</a>
%H A175655 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-6).
%F A175655 G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).
%F A175655 a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=4 and a(2)=8.
%F A175655 a(n) = ((10+8*A)*A^(-n-1) + (10+8*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
%F A175655 Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n)-A006130(n-1)*sqrt(13)).
%F A175655 E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 5*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - _Stefano Spezia_, Jan 31 2023
%p A175655 with(LinearAlgebra): nmax:=27; m:=5; 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);
%t A175655 CoefficientList[Series[(1 + x - 5 x^2) / (1 - 3 x - x^2 + 6 x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 21 2013 *)
%t A175655 LinearRecurrence[{3,1,-6},{1,4,8},40] (* _Harvey P. Dale_, Dec 25 2024 *)
%o A175655 (Magma) I:=[1, 4, 8]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-6*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jul 21 2013
%o A175655 (PARI) a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;4;8])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%Y A175655 Cf. Elephant sequences central square [decimal value A[5]]: A000007 [0], A000012 [16], A000045 [1], A011782 [2], A000079 [3], A003945 [42], A099036 [11], A175656 [7], A105476 [69], A168604 [26], A045891 [19], A078057 [21], A151821 [170], A175657 [43], 4*A172481 [15; n>=-1], A175655 [71, this sequence], 4*A026597 [325; n>=-1], A033484 [58], A087447 [27], A175658 [23], A026150 [85], A175661 [171], A036563 [186], A098156 [59], A046717 [341], 2*A001792 [187; n>=1 with a(0)=1], A175659 [343].
%Y A175655 Cf. A000244, A006130, A075118, A175654.
%K A175655 easy,nonn
%O A175655 0,2
%A A175655 _Johannes W. Meijer_, Aug 06 2010, Aug 10 2010