A253112 Number of (n+2)X(1+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, 1342, 4619, 14541, 34786, 113891, 233392, 525617, 853971, 2327441, 3609337, 6243433, 8189113, 18881723, 25291483, 37714007, 44777219, 90840301, 112309445, 153843349, 173399285, 320621751, 377531207, 490093227, 535756183

Offset

1,1

Comments

Column 1 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-4) -8*a(n-5) -28*a(n-8) +28*a(n-9) +56*a(n-12) -56*a(n-13) -70*a(n-16) +70*a(n-17) +56*a(n-20) -56*a(n-21) -28*a(n-24) +28*a(n-25) +8*a(n-28) -8*a(n-29) -a(n-32) +a(n-33) for n>43.

Empirical for n mod 4 = 0: a(n) = (1/2880)*n^8 + (1/35)*n^7 + (169/160)*n^6 + (91/12)*n^5 - (170437/960)*n^4 - (4291/60)*n^3 + (547067/45)*n^2 - (1847017/42)*n - 22525 for n>10.

Empirical for n mod 4 = 1: a(n) = (1/2880)*n^8 + (1/35)*n^7 + (1493/1440)*n^6 + (871/120)*n^5 - (506711/2880)*n^4 + (3067/30)*n^3 + (1371401/120)*n^2 - (11084513/210)*n + (371127/16) for n>10.

Empirical for n mod 4 = 2: a(n) = (1/2880)*n^8 + (13/504)*n^7 + (85/96)*n^6 + (2389/720)*n^5 - (50539/320)*n^4 + (93863/144)*n^3 + (142985/18)*n^2 - (26157409/420)*n + (399767/4) for n>10.

Empirical for n mod 4 = 3: a(n) = (1/2880)*n^8 + (79/2520)*n^7 + (187/160)*n^6 + (7409/720)*n^5 - (190777/960)*n^4 - (235699/720)*n^3 + (5757523/360)*n^2 - (52286503/840)*n + (146121/16) for n>10

Example

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

Crossrefs

Sequence in context: A140851 A337428 A253119 * A211146 A155700 A108878

Adjacent sequences: A253109 A253110 A253111 * A253113 A253114 A253115

Keyword

nonn

Author

R. H. Hardin, Dec 27 2014

Status

approved