A253417 Number of (n+2)X(1+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 1, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.

17, 56, 257, 642, 1581, 2389, 5716, 7691, 11429, 13229, 25870, 30605, 39701, 42941, 73200, 81815, 99709, 104837, 163706, 177321, 208349, 215813, 316972, 336707, 386101, 396349, 556166, 583141, 657029, 670509, 908040, 943375, 1048781, 1065941

Offset

1,1

Comments

Column 1 of A253424.

Links

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

Formula

Empirical: a(n) = a(n-1) +4*a(n-4) -4*a(n-5) -6*a(n-8) +6*a(n-9) +4*a(n-12) -4*a(n-13) -a(n-16) +a(n-17) for n>21.

Empirical for n mod 4 = 0: a(n) = (7/12)*n^4 + 9*n^3 + (563/12)*n^2 - (745/2)*n + 671 for n>4.

Empirical for n mod 4 = 1: a(n) = (7/12)*n^4 + 9*n^3 + (479/12)*n^2 - (643/2)*n + 701 for n>4.

Empirical for n mod 4 = 2: a(n) = (7/12)*n^4 + (20/3)*n^3 + (365/12)*n^2 - (1058/3)*n + 1214 for n>4.

Empirical for n mod 4 = 3: a(n) = (7/12)*n^4 + (34/3)*n^3 + (509/12)*n^2 - (928/3)*n + 515 for n>4.

Example

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

Crossrefs

Sequence in context: A309032 A352282 A290763 * A117390 A141841 A146682

Adjacent sequences: A253414 A253415 A253416 * A253418 A253419 A253420

Keyword

nonn

Author

R. H. Hardin, Dec 31 2014

Status

approved