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A086346 On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner square. 10
1, 3, 18, 80, 400, 1904, 9248, 44544, 215296, 1039104, 5018112, 24227840, 116985856, 564850688, 2727354368, 13168803840, 63584665600, 307013812224, 1482394042368, 7157631156224 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Johannes W. Meijer, Aug 01 2010: (Start)

The a(n) represent the number of n-move paths of a chess king on a 3 X 3 board that end or start in a given corner square m (m = 1, 3, 7, 9). To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the king's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.

Inverse binomial transform of A079291 (without the leading 0).

(End)

From R. J. Mathar, Oct 12 2010: (Start)

The row n=3 of an array counting king walks on an n X n board with k steps, starting from a corner:

1,3,9,27,81,243,729,2187,6561,19683,59049,177147,531441,

1,3,18,80,400,1904,9248,44544,215296,1039104,5018112,24227840,

1,3,18,105,615,3600,21075,123375,722250,4228125,24751875,144900000,

1,3,18,105,684,4359,28278,182349,1179792,7622667,49283802,318543921,

1,3,18,105,684,4550,30807,209867,1434279,9815190,67209723,460343730,

1,3,18,105,684,4550,31340,218056,1533712,10829360,76720288,544266448,

1,3,18,105,684,4550,31340,219555,1559835,11177190,80573373,583082082,

1,3,18,105,684,4550,31340,219555,1564080,11259785,81765550,597274600,

1,3,18,105,684,4550,31340,219555,1564080,11271876,82025163,601303692,

1,3,18,105,684,4550,31340,219555,1564080,11271876,82059768,602116173,

1,3,18,105,684,4550,31340,219555,1564080,11271876,82059768,602215614,

The partial sums along the rows are documented in A123109 (king walks with between 1 and k steps). (End)

REFERENCES

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984. [From Johannes W. Meijer, Aug 01 2010]

LINKS

Table of n, a(n) for n=0..19.

Mike Oakes, KingMovesForPrimes.

Zak Seidov et al., New puzzle? King moves for primes, digest of 28 messages in primenumbers group, Jul 13 - Jul 23, 2003. [Cached copy]

Zak Seidov, KingMovesForPrimes.

Sleephound, KingMovesForPrimes.

Index entries for linear recurrences with constant coefficients, signature (2,12,8). [From R. J. Mathar, Jul 22 2010]

FORMULA

a(n) = (1/32)(2(-2)^(n+2) + (2+sqrt(8))^(n+2) + (2-sqrt(8))^(n+2)).

From R. J. Mathar, Jul 22 2010: (Start)

a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3).

G.f.: ( -1-x ) / ( (2*x+1)*(4*x^2+4*x-1) ).

a(n) = A057087(n-1)/2 + 3*A057087(n)/4 + (-2)^n/4. (End)

Lim_{k->infinity} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2). - Johannes W. Meijer, Aug 01 2010

MAPLE

with(LinearAlgebra): nmax:=19; m:=1; A[5]:= [1, 1, 1, 1, 0, 1, 1, 1, 1]: A:=Matrix([[0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0, 0], [0, 1, 0, 0, 1, 1, 0, 0, 0], [1, 1, 0, 0, 1, 0, 1, 1, 0], A[5], [0, 1, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 0, 0, 1, 0], [0, 0, 0, 1, 1, 1, 1, 0, 1], [0, 0, 0, 0, 1, 1, 0, 1, 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); # Johannes W. Meijer, Aug 01 2010

MATHEMATICA

Table[(1/32)(2(-2)^(n+2)+(2+Sqrt[8])^(n+2)+(2-Sqrt[8])^(n+2)), {n, 0, 19}]

CROSSREFS

Cf. A086347, A086348, A086349; Cf. A179596. - Johannes W. Meijer, Aug 01 2010

Sequence in context: A308853 A309257 A135371 * A036290 A078904 A099012

Adjacent sequences:  A086343 A086344 A086345 * A086347 A086348 A086349

KEYWORD

nonn,changed

AUTHOR

Zak Seidov, Jul 17 2003

EXTENSIONS

Offset changed and edited by Johannes W. Meijer, Jul 15 2010

STATUS

approved

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Last modified November 22 06:15 EST 2019. Contains 329389 sequences. (Running on oeis4.)