A253113 Number of (n+2)X(2+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.

272, 920, 6334, 28600, 139760, 502272, 2259097, 5670421, 21399557, 39120366, 120865988, 183106294, 479657064, 649963111, 1500762543, 1895383751, 3973980189, 4787211558, 9308563998, 10844855216, 19852566186, 22566185885

Offset

1,1

Comments

Column 2 of A253119.

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Formula

Empirical: a(n) = a(n-1) +8*a(n-2) -8*a(n-3) -28*a(n-4) +28*a(n-5) +56*a(n-6) -56*a(n-7) -70*a(n-8) +70*a(n-9) +56*a(n-10) -56*a(n-11) -28*a(n-12) +28*a(n-13) +8*a(n-14) -8*a(n-15) -a(n-16) +a(n-17) for n>29.

Empirical for n mod 2 = 0: a(n) = (8/45)*n^8 + (1504/315)*n^7 + (3571/45)*n^6 - (84011/45)*n^5 + (5095469/1440)*n^4 + (27027629/360)*n^3 - (17188069/180)*n^2 - (433790313/140)*n + 11000595 for n>12.

Empirical for n mod 2 = 1: a(n) = (8/45)*n^8 + (1952/315)*n^7 + (301/3)*n^6 - (70897/45)*n^5 - (1084059/160)*n^4 + (23349461/180)*n^3 - (6792595/144)*n^2 - (509001211/140)*n + (327336053/32) for n>12.

Example

Some solutions for n=4:
..0..2..2..3....0..2..2..3....0..2..2..3....0..2..1..3....0..2..2..3
..2..2..1..2....3..3..1..1....2..3..1..1....1..2..1..1....2..3..1..2
..2..1..3..2....2..1..3..2....2..1..3..2....2..1..2..2....1..1..2..3
..2..2..2..2....2..2..2..1....2..1..2..2....2..1..2..2....2..2..2..1
..2..2..2..2....1..2..2..2....2..2..1..2....1..2..2..1....2..2..2..2
..2..2..3..3....3..2..2..3....2..2..2..3....2..2..2..2....2..2..2..3
Knight distance matrix for n=4:
..0..3..2..5
..3..4..1..2
..2..1..4..3
..3..2..3..2
..2..3..2..3
..3..4..3..4

Crossrefs

Sequence in context: A316374 A306134 A317269 * A234884 A158587 A304417

Adjacent sequences: A253110 A253111 A253112 * A253114 A253115 A253116

Keyword

nonn

Author

R. H. Hardin, Dec 27 2014

Status

approved