A180319 Total number of possible standard knight moves on an n X 2n chessboard, if the knight is placed anywhere

0, 8, 52, 128, 236, 376, 548, 752, 988, 1256, 1556, 1888, 2252, 2648, 3076, 3536, 4028, 4552, 5108, 5696, 6316, 6968, 7652, 8368, 9116, 9896, 10708, 11552, 12428, 13336, 14276, 15248, 16252, 17288, 18356, 19456, 20588, 21752, 22948, 24176, 25436

Offset

1,2

Comments

a(n) counts every possible moves of a standard chess knight placed anywhere.

For examples, in usual chessboard 8X8 a knight in a corner has only 2 moves,

in a central square it has the maximum number of moves:8.

Summing all over the 64 squares we have 336 possible moves.

Instead on a chessboard 4x8 the number is reduced:

-----------------

|2|3|4|4|4|4|3|2|

-----------------

|3|4|6|6|6|6|4|3|

-----------------

|3|4|6|6|6|6|4|3|

-----------------

|2|3|4|4|4|4|3|2|

-----------------

the total number is 128

Links

Table of n, a(n) for n=1..41.

Formula

Conjecture: a(n) = 4*(4-9*n+4*n^2) for n>1. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>4. G.f.: 4*x^2*(2+7*x-x^2)/(1-x)^3. [Colin Barker, Mar 11 2012]

Crossrefs

Cf. A035008

Sequence in context: A238648 A341626 A215834 * A199706 A302318 A035288

Adjacent sequences: A180316 A180317 A180318 * A180320 A180321 A180322

Keyword

easy,nonn

Author

Graziano Aglietti (mg5055(AT)mclink.it), Aug 27 2010

Status

approved