A253004 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 3 and every value within 3 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 4, 1, 0, 1, 4, 10, 14, 1, 1, 14, 10, 20, 55, 34, 1, 34, 55, 20, 35, 140, 279, 69, 69, 279, 140, 35, 56, 285, 1028, 1132, 69, 1132, 1028, 285, 56, 84, 506, 2601, 7235, 3072, 3072, 7235, 2601, 506, 84, 120, 819, 5318, 25233, 39758, 3072

Offset

1,11

Links

R. H. Hardin, Table of n, a(n) for n = 1..1200

Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 21.

Robert Dougherty-Bliss and Manuel Kauers, Hardinian Arrays, arXiv:2309.00487 [math.CO], 2023. Hardinian Arrays, El. J. Combinat. 31 (2) (2024) #P2.9

Formula

Empirical for column k:

k=1: a(n) = (1/6)*n^3 - 1*n^2 + (11/6)*n - 1.

k=2: a(n) = (8/3)*n^3 - 26*n^2 + (253/3)*n - 91 for n>2.

k=3: a(n) = (160/3)*n^3 - 708*n^2 + (9539/3)*n - 4831 for n>4.

k=4: a(n) = (4096/3)*n^3 - 22816*n^2 + (388490/3)*n - 249567 for n>6.

k=5: a(n) = (133120/3)*n^3 - 893616*n^2 + (18332582/3)*n - 14187577 for n>8.

k=6: a(n) = (5242880/3)*n^3 - 41275392*n^2 + (991610656/3)*n - 897487301 for n>10.

k=7: a(n) = (235012096/3)*n^3 - 2126491008*n^2 + (58625640404/3)*n - 60801081325 for n>12.

Example

Table starts:
..0...0....0......1.......4.......10........20.........35.........56.........84
..0...0....0......1......14.......55.......140........285........506........819
..0...0....0......1......34......279......1028.......2601.......5318.......9499
..1...1....1......1......69.....1132......7235......25233......63135.....129133
..4..14...34.....69......69.....3072.....39758.....228484.....775433....1932763
.10..55..279...1132....3072.....3072....122833....1486152....8270017...27983105
.20.140.1028...7235...39758...122833....122833....4915726...59154789..329035981
.35.285.2601..25233..228484..1486152...4915726....4915726..204051186.2492354946
.56.506.5318..63135..775433..8270017..59154789..204051186..204051186.8849413857
.84.819.9499.129133.1932763.27983105.329035981.2492354946.8849413857.8849413857
Some solutions for n=6 and k=4:
..0..0..1..2....0..0..1..2....0..0..1..2....0..0..1..1....0..0..1..1
..0..0..1..2....1..1..1..2....0..0..1..2....0..0..1..1....0..1..1..1
..1..1..1..2....1..1..2..2....0..0..1..2....0..1..1..2....1..1..1..1
..1..1..2..2....2..2..2..2....0..1..1..2....1..1..1..2....1..1..1..2
..2..2..2..2....2..2..2..2....1..1..1..2....1..1..1..2....1..2..2..2
..2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2

Crossrefs

Column 1 is A000292(n-3).

Column 2 is A100157(n-3).

Sequence in context: A290457 A344851 A346202 * A253011 A290456 A010639

Adjacent sequences: A253001 A253002 A253003 * A253005 A253006 A253007

Keyword

nonn,tabl,changed

Author

R. H. Hardin, Dec 25 2014

Status

approved