A180413 Total number of possible knight moves on an n X n X n chessboard, if the knight is placed anywhere.

0, 144, 576, 1440, 2880, 5040, 8064, 12096, 17280, 23760, 31680, 41184, 52416, 65520, 80640, 97920, 117504, 139536, 164160, 191520, 221760, 255024, 291456, 331200, 374400, 421200, 471744, 526176, 584640, 647280, 714240, 785664, 861696

Offset: 1

Author

Graziano Aglietti (mg5055(AT)mclink.it), Sep 02 2010

Keywords

• easy
• nice
• nonn

Comments

The maximum number of move in tridimensional chessboard is 24, 8 for every dimension. In a vertex the number is smaller.

Binomial transform of [144, 432, 432, 144, 0, 0, 0, ...] = (144, 576, 1440, ...). - Gary W. Adamson, Sep 03 2010

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A027480, A135503, A180319, A035008.

Programs

Mathematica

Table[24n(n^2-1),{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,144,576,1440},40] (* Harvey P. Dale, Feb 13 2013 *)

PARI

a(n)=24*n*(n^2-1) \\ Charles R Greathouse IV, Nov 03 2014

Formula

a(n) = 24*n*(n^2-1).

G.f.: 144*x^2/(1-x)^4. - Colin Barker, Mar 17 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=0, a(2)=144, a(3)=576, a(4)=1440. - Harvey P. Dale, Feb 13 2013

E.g.f.: 24 * exp(x) * x^2 * (3 + x). - Vaclav Kotesovec, Feb 15 2015