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A086348 On a 3 X 3 board, number of n-move routes of chess king ending in the central square. 8
1, 8, 32, 168, 784, 3840, 18432, 89216, 430336, 2078720, 10035200, 48457728, 233967616, 1129709568, 5454692352, 26337640448, 127169265664, 614027755520, 2964787822592, 14315262836736 (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 the central square m (m = 5).

Inverse binomial transform of A090390 (without the first leading 1).

(End)

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

The row n=3 of an array T(n,k) counting king walks on an n X n board starting on a square on the diagonal next to a corner:

1,8,32,168,784,3840,18432,89216,430336,2078720,10035200,48457728,233967616,

1,8,47,275,1610,9425,55175,323000,1890875,11069375,64801250,379353125,

1,8,47,318,2013,13140,84555,547722,3537081,22874400,147831399,955690326,

1,8,47,318,2134,14539,99267,679189,4650100,31848677,218164072,1494530576,

1,8,47,318,2134,14880,103920,733712,5187856,36796224,261164848,1855327584,

1,8,47,318,2134,14880,104885,748845,5382180,38880243,281743740,2045995632,

1,8,47,318,2134,14880,104885,751590,5430735,39556080,289541500,2127935700,

1,8,47,318,2134,14880,104885,751590,5438580,39710495,291852880,2156410817,

1,8,47,318,2134,14880,104885,751590,5438580,39733008,292340803,2164218694,

1,8,47,318,2134,14880,104885,751590,5438580,39733008,292405638,2165752797, (End)

LINKS

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

Mike Oakes, KingMovesForPrimes.

Zak Seidov, 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]

Sleephound, KingMovesForPrimes.

FORMULA

a(n) = (1/16)(4(-2)^(n+1) + (2+sqrt(8))^(n+2) + (2-sqrt(8))^(n+2)).

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

G.f.: ( 1+6*x+4*x^2 ) / ( (2*x+1)*(-4*x^2-4*x+1) ).

a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3) with a(0)=1, a(1)=8 and a(2)=32.

Lim_{k->infinity} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2). (End)

2*a(n) = 3*A057087(n) + 2*A057087(n-1) - (-2)^n. - R. J. Mathar, May 21 2019

MAPLE

with(LinearAlgebra): nmax:=19; m:=5; 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[(1/16)(4(-2)^(n+1)+(2+Sqrt[8])^(n+2)+(2-Sqrt[8])^(n+2)), {n, 0, 19}]

CROSSREFS

Cf. A086346, A086347.

Cf. A179597.

Sequence in context: A188121 A045684 A045675 * A129798 A129792 A224664

Adjacent sequences:  A086345 A086346 A086347 * A086349 A086350 A086351

KEYWORD

nonn,easy

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 August 6 19:52 EDT 2020. Contains 336256 sequences. (Running on oeis4.)