A179607 -id:A179607 - cp's OEIS frontend

A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5x + 2x^2)/(1 - 2x - 11x^2 - 6*x^3).

1, 7, 27, 137, 613, 2895, 13355, 62233, 288741, 1342175, 6233899, 28964169, 134554277, 625117807, 2904117675, 13491856889, 62679715045, 291194561919, 1352817130667, 6284852732713, 29197861274277, 135646005392399

Offset: 0

Johannes W. Meijer, Jul 28 2010, Aug 10 2010

The a(n) represent 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 king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
For the central square the 512 red kings lead to 47 different red king sequences, see the cross-references for some examples.
The sequence above corresponds to four A[5] vectors with decimal [binary] values 367 [1,0,1,1,0,1,1,1,1], 463 [1,1,1,0,0,1,1,1,1], 487 [1,1,1,1,0,0,1,1,1] and 493 [1,1,1,1,0,1,1,0,1]. These vectors lead for the corner squares to A179596 and for the side squares to A126473.
This sequence belongs to a family of sequences with g.f. (1 + (k+2)*x + (2*k-4)*x^2)/(1 - 2*x - (k+8)*x^2 - (2*k)*x^3). Red king sequences that are members of this family are A179607 (k=0), A179605 (k=1), A179601 (k=2), A179597 (k=3; this sequence) and A086348 (k=4). Another member of this family is A179609 (k = -4).

Index entries for linear recurrences with constant coefficients, signature (2,11,6).

Red king sequences central square [numerical value A[5]]: A086348 [495], A179599 [239], A179597 [367], A179601 [335], A179603 [95], A154964 [31], A179605 [327], A179606 [27], A179611 [15], A179607 [325], A015521 [11], A007483 [2], A000012 [16], A000007 [0].

with(LinearAlgebra): nmax:=21; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [1,0,1,1,0,1,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: 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);

LinearRecurrence[{2,11,6},{1,7,27},30] (* Harvey P. Dale, Mar 01 2015 *)

G.f.: (1 + 5*x + 2*x^2)/(1 - 2*x - 11*x^2 - 6*x^3).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) with a(0) = 1, a(1) = 7 and a(2) = 27.
a(n) = 8*(-1/2)^(-n+1)/9 + ((7+11*sqrt(7))*A^(-n-1) + (7-11*sqrt(7))*B^(-n-1))/126 with A = (-2+sqrt(7))/3 and B = (-2-sqrt(7))/3.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*(A000244(n)/(A015530(n)*sqrt(7) - A108851(n))).

A100284 Expansion of (1-4x-x^2)/((1-x)(1-4x-5x^2)).

Original entry on oeis.org

1, 1, 5, 21, 105, 521, 2605, 13021, 65105, 325521, 1627605, 8138021, 40690105, 203450521, 1017252605, 5086263021, 25431315105, 127156575521, 635782877605, 3178914388021, 15894571940105, 79472859700521, 397364298502605

Offset: 0

Paul Barry, Nov 11 2004

Binomial transform of A054881.

Binomial transform of A179607. - Johannes W. Meijer, Aug 01 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,1,-5)

Cf. A054881, A100285, A100286, A179607.

[(5^n +2*(-1)^n +3)/6: n in [0..40]]; // G. C. Greubel, Feb 06 2023

CoefficientList[Series[(1-4x-x^2)/((1-x)(1-4x-5x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{5,1,-5},{1,1,5},30] (* Harvey P. Dale, Apr 01 2013 *)

def A100284(n): return (1/6)*(5^n +1 +4*((n+1)%2))
[A100284(n) for n in range(41)] # G. C. Greubel, Feb 06 2023

a(n) = 5*a(n-1) + a(n-2) - 5*a(n-3).
a(n) = (1/6)*(3 + 5^n + 2*(-1)^n).
E.g.f.: (1/6)*(exp(5*x) + 3*exp(x) + 2*exp(-x)). - G. C. Greubel, Feb 06 2023

cp's OEIS Frontend

A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5x + 2x^2)/(1 - 2x - 11x^2 - 6*x^3).

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A100284 Expansion of (1-4x-x^2)/((1-x)(1-4x-5x^2)).

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cp's OEIS Frontend

A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5*x + 2*x^2)/(1 - 2*x - 11*x^2 - 6*x^3).

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Crossrefs

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Maple

Mathematica

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A100284 Expansion of (1-4*x-x^2)/((1-x)*(1-4*x-5*x^2)).

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Magma

Mathematica

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Formula

A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5x + 2x^2)/(1 - 2x - 11x^2 - 6*x^3).

A100284 Expansion of (1-4x-x^2)/((1-x)(1-4x-5x^2)).