A253114 Number of (n+2)X(3+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.

1342, 6334, 64228, 426571, 2801464, 11529235, 79249122, 269081698, 1126140254, 2033916546, 10398959008, 19253327106, 49068963720, 62663185107, 216434297394, 306964534797, 600045409295, 683103473164, 1832436683833, 2296747272574

Offset

1,1

Comments

Column 3 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>48.

Empirical for n mod 4 = 0: a(n) = (48128/315)*n^8 - (70784/45)*n^7 + (1613986/45)*n^6 - (408827287/360)*n^5 + (656806290529/46080)*n^4 - (417508867243/5760)*n^3 + (1340641181597/20160)*n^2 + (68326310189/120)*n - 1314231925 for n>15.

Empirical for n mod 4 = 1: a(n) = (48128/315)*n^8 - (399232/315)*n^7 + (494198/15)*n^6 - (391895399/360)*n^5 + (200444821003/15360)*n^4 - (719258439109/11520)*n^3 + (8134684965509/161280)*n^2 + (12687388527967/26880)*n - (1104560946555/1024) for n>15.

Empirical for n mod 4 = 2: a(n) = (48128/315)*n^8 - (784256/315)*n^7 + (254546/5)*n^6 - (494044999/360)*n^5 + (99930960601/5120)*n^4 - (787756612561/5760)*n^3 + (17328260184551/40320)*n^2 - (780223082251/3360)*n - (71148317231/64) for n>15.

Empirical for n mod 4 = 3: a(n) = (48128/315)*n^8 - (110464/315)*n^7 + (516898/45)*n^6 - (54346123/72)*n^5 + (321821523169/46080)*n^4 + (153162748967/11520)*n^3 - (67343538926807/161280)*n^2 + (8549722002355/5376)*n - (1254021550131/1024) for n>15.

Example

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

Crossrefs

Sequence in context: A043647 A023064 A252266 * A237506 A264385 A167437

Adjacent sequences: A253111 A253112 A253113 * A253115 A253116 A253117

Keyword

nonn

Author

R. H. Hardin, Dec 27 2014

Status

approved