A253335 Number of (n+2)X(1+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 3, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.

69, 488, 1928, 7494, 27015, 87621, 319172, 945613, 2874539, 6935762, 22983454, 49969332, 119548267, 214145613, 622017082, 1069718995, 2070309209, 3028231177, 7700984072, 11401562937, 19017598555, 24702675207, 56114922678, 75683326471

Offset

1,1

Comments

Column 1 of A253342.

Links

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

Formula

Empirical: a(n) = a(n-1) +12*a(n-4) -12*a(n-5) -66*a(n-8) +66*a(n-9) +220*a(n-12) -220*a(n-13) -495*a(n-16) +495*a(n-17) +792*a(n-20) -792*a(n-21) -924*a(n-24) +924*a(n-25) +792*a(n-28) -792*a(n-29) -495*a(n-32) +495*a(n-33) +220*a(n-36) -220*a(n-37) -66*a(n-40) +66*a(n-41) +12*a(n-44) -12*a(n-45) -a(n-48) +a(n-49) for n>66

Empirical for n mod 4 = 0: a(n) = (1/34214400)*n^12 + (19/3326400)*n^11 + (11413/21772800)*n^10 + (2543/120960)*n^9 + (171211/1036800)*n^8 - (17681/1800)*n^7 - (364775123/3110400)*n^6 + (503575579/120960)*n^5 - (23104979617/5443200)*n^4 - (42603595499/75600)*n^3 + (271867785329/59400)*n^2 - (1488535171/330)*n - 43423533 for n>17

Empirical for n mod 4 = 1: a(n) = (1/34214400)*n^12 + (19/3326400)*n^11 + (11329/21772800)*n^10 + (139/6720)*n^9 + (1113277/7257600)*n^8 - (80513/8400)*n^7 - (321868919/3110400)*n^6 + (24501551/6048)*n^5 - (2717152987/340200)*n^4 - (13345114753/25200)*n^3 + (32592081480029/6652800)*n^2 - (327571973557/36960)*n - (2047419033/64) for n>17

Empirical for n mod 4 = 2: a(n) = (1/34214400)*n^12 + (107/19958400)*n^11 + (1453/3110400)*n^10 + (12203/725760)*n^9 + (59611/1036800)*n^8 - (2635259/302400)*n^7 - (138251441/3110400)*n^6 + (2772874619/725760)*n^5 - (630113333/24300)*n^4 - (150767030627/453600)*n^3 + (1190430331397/237600)*n^2 - (972074050223/55440)*n - (25910239/4) for n>17

Empirical for n mod 4 = 3: a(n) = (1/34214400)*n^12 + (11/1814400)*n^11 + (12403/21772800)*n^10 + (3487/145152)*n^9 + (1620937/7257600)*n^8 - (1082489/100800)*n^7 - (453300833/3110400)*n^6 + (3176992123/725760)*n^5 + (6382948247/1360800)*n^4 - (646363851967/907200)*n^3 + (35282024010953/6652800)*n^2 - (80650610293/20160)*n - (3453848023/64) for n>17

Example

Some solutions for n=4
..0..2..1....0..3..2....0..1..1....0..1..1....0..1..0....0..2..1....0..2..2
..2..2..1....2..4..1....2..2..1....2..2..0....1..2..0....1..3..0....2..1..1
..1..1..2....2..1..3....1..1..2....1..0..2....0..0..1....1..1..2....1..1..2
..2..1..2....3..2..3....2..2..2....1..1..1....1..0..1....2..1..2....1..1..1
..1..1..1....2..2..2....1..2..2....0..2..0....0..1..0....1..2..1....1..2..1
..2..3..2....3..3..2....3..2..2....1..2..1....0..2..0....2..2..2....2..1..1
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: A161486 A236158 A253342 * A234832 A234825 A251010

Adjacent sequences: A253332 A253333 A253334 * A253336 A253337 A253338

Keyword

nonn

Author

R. H. Hardin, Dec 30 2014

Status

approved