|
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), 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)
Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 12 2010: (Start)
The row n=3 of an array T(n,k) counting king walks on an n-by-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
| Mike Oakes, KingMovesForPrimes.
Zak Seidov, KingMovesForPrimes.
Sleephound, KingMovesForPrimes.
|
|
|
FORMULA
| a(n)=(1/16)(4(-2)^(n+1)+(2+Sqrt[8])^(n+2)+(2-Sqrt[8])^(n+2))
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 01 2010: (Start)
GF(x) = (1+6*x+4*x^2)/(1-2*x-12*x^2-8*x^3)
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.
Limit(a(n+k)/a(k),k=infinity) = A084128(n) + 2*A057087(n-1)*sqrt(2)
(End)
|
|
|
MAPLE
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 01 2010: (Start)
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);
(End)
|
|
|
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.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 01 2010: (Start)
Cf. A179597.
(End)
Sequence in context: A188121 A045684 A045675 * A129798 A129792 A093759
Adjacent sequences: A086345 A086346 A086347 * A086349 A086350 A086351
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Zak Seidov (zakseidov(AT)yahoo.com), Jul 17 2003
|
|
|
EXTENSIONS
| Offset changed and edited by Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 15 2010
|
| |
|
|
