A253119 T(n,k)=Number of (n+2)X(k+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 2, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.

53, 272, 272, 1342, 920, 1342, 4619, 6334, 6334, 4619, 14541, 28600, 64228, 28600, 14541, 34786, 139760, 426571, 426571, 139760, 34786, 113891, 502272, 2801464, 3950239, 2801464, 502272, 113891, 233392, 2259097, 11529235, 28256241, 28256241

Offset

1,1

Comments

Table starts

.....53......272.......1342.........4619.........14541...........34786

....272......920.......6334........28600........139760..........502272

...1342.....6334......64228.......426571.......2801464........11529235

...4619....28600.....426571......3950239......28256241.......201911670

..14541...139760....2801464.....28256241.....343085240......3518586637

..34786...502272...11529235....201911670....3518586637.....40022802662

.113891..2259097...79249122...1873418602...32095583880....638441329329

.233392..5670421..269081698...7421334529..235545304374...7166392767013

.525617.21399557.1126140254..54446797689.2365327348752..68323597331510

.853971.39120366.2033916546.133685068828.6157835478824.307062940433405

Links

R. H. Hardin, Table of n, a(n) for n = 1..241

Formula

Empirical for column k:

k=1: [linear recurrence of order 33] for n>43

k=2: [order 17] for n>29

k=3: [same order 33] for n>48

k=4: [same order 17] for n>37

k=5: [same order 33] for n>63

k=6: [same order 17] for n>65

Empirical quasipolynomials for column k:

k=1: polynomial of degree 8 plus a quasipolynomial of degree 7 with period 4 for n>10

k=2: polynomial of degree 8 plus a quasipolynomial of degree 7 with period 2 for n>12

k=3: polynomial of degree 8 plus a quasipolynomial of degree 7 with period 4 for n>15

k=4: polynomial of degree 8 plus a quasipolynomial of degree 7 with period 2 for n>20

k=5: polynomial of degree 8 plus a quasipolynomial of degree 7 with period 4 for n>30

k=6: polynomial of degree 8 plus a quasipolynomial of degree 7 with period 2 for n>48

Example

Some solutions for n=3 k=4:
..0..1..1..2..1..1....0..1..1..2..1..2....0..1..1..2..1..2....0..1..1..2..1..2
..1..2..0..1..2..2....1..2..0..1..2..2....1..2..0..1..1..2....1..2..0..1..2..2
..1..0..2..1..1..1....1..0..2..2..1..1....1..0..2..1..1..1....1..0..2..1..1..2
..1..0..1..1..1..2....1..1..1..1..1..2....1..1..2..1..2..2....2..0..1..1..2..2
..1..2..1..1..2..2....0..2..1..2..2..1....1..1..1..2..2..1....1..1..1..1..2..1
Knight distance matrix for n=3:
..0..3..2..3..2..3
..3..4..1..2..3..4
..2..1..4..3..2..3
..3..2..3..2..3..4
..2..3..2..3..4..3

Crossrefs

Sequence in context: A241488 A140851 A337428 * A253112 A211146 A155700

Adjacent sequences: A253116 A253117 A253118 * A253120 A253121 A253122

Keyword

nonn,tabl

Author

R. H. Hardin, Dec 27 2014

Status

approved