A086346 On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner square.

1, 3, 18, 80, 400, 1904, 9248, 44544, 215296, 1039104, 5018112, 24227840, 116985856, 564850688, 2727354368, 13168803840, 63584665600, 307013812224, 1482394042368, 7157631156224, 34560101318656, 166870928850944, 805724122775552, 3890380202311680, 18784417308737536, 90699190027419648

Offset: 0

Author

Zak Seidov, Jul 17 2003

Keywords

• nonn
• easy

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, ...;

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

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

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

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

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

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

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

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

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

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

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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).

Crossrefs

Cf. A086347, A086348, A086349.

Cf. A179596. - Johannes W. Meijer, Aug 01 2010

Cf. A002203, A057087, A079291, A084128, A110048, A123109.

Programs

Magma

[2^(n-3)*(Evaluate(DicksonFirst(n+2,-1), 2) +2*(-1)^n): n in [0..30]]; // G. C. Greubel, Aug 18 2022

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}] // FullSimplify
LinearRecurrence[{2,12,8}, {1,3,18}, 31] (* G. C. Greubel, Aug 18 2022 *)

PARI

Vec((1+x)/((1+2*x)*(1-4*x-4*x^2))+O(x^30)) \\ Joerg Arndt, Jan 29 2024

SageMath

[2^(n-3)*(lucas_number2(n+2,2,-1) +2*(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 18 2022

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

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

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

a(n) = A110048(n) + A110048(n-1). - R. J. Mathar, Mar 08 2021

a(n) = 2^(n-3)*(A002203(n+2) + 2*(-1)^n). - G. C. Greubel, Aug 18 2022

Extensions

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