A083424 -id:A083424 - cp's OEIS frontend

A179607 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x - 4x^2)/(1 - 2x - 8x^2).

1, 4, 12, 56, 208, 864, 3392, 13696, 54528, 218624, 873472, 3495936, 13979648, 55926784, 223690752, 894795776, 3579117568, 14316601344, 57266143232, 229065097216, 916259340288, 3665039458304, 14660153638912, 58640622944256

Offset: 0

Johannes W. Meijer, Jul 28 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.
The sequence above corresponds to just one red king vector, i.e., A[5] vector, with decimal [binary] value 325 [1,0,1,0,0,0,1,0,1]. This vectors leads for the corner squares to A083424 and for the side squares to A003947.
The inverse binomial transform of A100284 (without the first leading 1).

Index entries for linear recurrences with constant coefficients, signature (2, 8).

Cf. A179597 (central square).

with(LinearAlgebra): nmax:=24; 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]:= [1,0,1,0,0,0,1,0,1]: 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);

Join[{1},LinearRecurrence[{2,8},{4,12},30]] (* Harvey P. Dale, Mar 01 2012 *)

G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).
a(n) = 2*a(n-1) + 8*a(n-2), for n >= 3, with a(0) = 1, a(1) = 4 and a(2) = 12.
a(n) = 5*(4)^(n)/6 - (-2)^(n)/3 for n >= 1 and a(0) = 1.
a(n) = 4*A083424(n-1), n>0. - R. J. Mathar, Mar 08 2021

A245181 Numbers appearing in A245180.

Original entry on oeis.org

1, 3, 8, 14, 24, 52, 64, 72, 112, 192, 216, 336, 416, 512, 576, 848, 896, 1248, 1536, 1568, 1728, 2688, 3328, 3424, 4096, 4608, 5184, 5824, 6784, 7168, 8064, 9984, 12288, 12544, 13632, 13824, 20352, 21504, 21632, 24192, 26624, 27392, 29952, 32768, 36864, 37632, 41472, 46592, 54272, 54656

Offset: 1

N. J. A. Sloane, Jul 17 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..325

Cf. A245180. A083424 is a subsequence.

A083423 a(n) = (5*3^n + (-3)^n)/6.

Original entry on oeis.org

1, 2, 9, 18, 81, 162, 729, 1458, 6561, 13122, 59049, 118098, 531441, 1062882, 4782969, 9565938, 43046721, 86093442, 387420489, 774840978, 3486784401, 6973568802, 31381059609, 62762119218, 282429536481, 564859072962

Offset: 0

Paul Barry, Apr 30 2003

Index entries for linear recurrences with constant coefficients, signature (0,9).

Cf. A083424.

LinearRecurrence[{0,9},{1,2},30] (* Harvey P. Dale, Dec 04 2019 *)

From Reinhard Zumkeller, Mar 04 2011: (Start)
G.f.: (1+2x)/((1-3x)*(1+3x)).
E.g.f.: (5*exp(3x) + exp(-3x))/6.
a(n+3) = a(n+2)*a(n+1)/a(n). (End)
a(n) = 9*a(n-2). - Vincenzo Librandi, Mar 20 2011

A122910 Expansion of (1-2x-3x^2)/((1-2x)(1+4x)(1-8x)).

Original entry on oeis.org

1, 4, 45, 302, 2636, 20184, 165040, 1305952, 10504896, 83809664, 671394560, 5367485952, 42954566656, 343577810944, 2748857364480, 21989919383552, 175923113148416, 1407369872769024, 11259019111628800, 90071912374730752

Offset: 0

Paul Barry, Sep 18 2006

Let M be the matrix M(n,k)=J(k+1)*sum{j=0..n, (-1)^(n-j)C(n,j)C(j+1,k+1)}. a(n) gives the row sums of M^3.

Index entries for linear recurrences with constant coefficients, signature (6,24,-64).

CoefficientList[Series[(1-2x-3x^2)/((1-2x)(1+4x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[{6, 24, -64}, {1, 4, 45}, 30] (* Harvey P. Dale, Jun 21 2011 *)

G.f.: (1-2x-3x^2)/(1-6x-24x^2+64x^3); a(n)=5*8^n/8+7*(-4)^n/24+2^n/12; a(n)=J(n)*A083424(n-1)+J(n+1)*A083424(n) where J(n) are the Jacobsthal numbers A001045(n).

a(0)=1, a(1)=4, a(2)=45, a(n)=6*a(n-1)+24*a(n-2)-64*a(n-3). - Harvey P. Dale, Jun 21 2011

cp's OEIS Frontend

A179607 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x - 4x^2)/(1 - 2x - 8x^2).

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A245181 Numbers appearing in A245180.

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A083423 a(n) = (5*3^n + (-3)^n)/6.

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A122910 Expansion of (1-2x-3x^2)/((1-2x)(1+4x)(1-8x)).

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cp's OEIS Frontend

A179607 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).

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Crossrefs

Programs

Maple

Mathematica

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A245181 Numbers appearing in A245180.

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Author

Keywords

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Crossrefs

A083423 a(n) = (5*3^n + (-3)^n)/6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A122910 Expansion of (1-2x-3x^2)/((1-2x)(1+4x)(1-8x)).

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Author

Keywords

Comments

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Programs

Mathematica

Formula

A179607 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x - 4x^2)/(1 - 2x - 8x^2).