A252938 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 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

1, 2, 2, 4, 5, 4, 8, 13, 13, 8, 15, 34, 44, 34, 15, 26, 83, 153, 153, 83, 26, 42, 176, 494, 711, 494, 176, 42, 64, 329, 1343, 3067, 3067, 1343, 329, 64, 93, 558, 3016, 10920, 17962, 10920, 3016, 558, 93, 130, 879, 5833, 30818, 86488, 86488, 30818, 5833, 879, 130, 176

Offset

1,2

Comments

Table starts

...1....2.....4......8......15.......26.........42..........64...........93

...2....5....13.....34......83......176........329.........558..........879

...4...13....44....153.....494.....1343.......3016........5833........10114

...8...34...153....711....3067....10920......30818.......70640.......138558

..15...83...494...3067...17962....86488.....320270......917811......2127013

..26..176..1343..10920...86488...578342....2952734....11219797.....32649081

..42..329..3016..30818..320270..2952734...21312696...113154831....440052087

..64..558..5833..70640..917811.11219797..113154831...857248091...4687944300

..93..879.10114.138558.2127013.32649081..440052087..4687944300..36723156004

.130.1308.16179.242764.4211511.76641323.1302939451.18615501830.205553855458

Links

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

Formula

Empirical for column k:

k=1: a(n) = (1/6)*n^3 - (1/2)*n^2 + (4/3)*n

k=2: a(n) = (8/3)*n^3 - 18*n^2 + (145/3)*n - 42 for n>2

k=3: a(n) = (160/3)*n^3 - 548*n^2 + (6071/3)*n - 2591 for n>4

k=4: a(n) = (4096/3)*n^3 - 18720*n^2 + (269642/3)*n - 149376 for n>6

k=5: a(n) = (133120/3)*n^3 - 760496*n^2 + (13526246/3)*n - 9199709 for n>8

k=6: [polynomial of degree 3] for n>10

k=7: [polynomial of degree 3] for n>12

Example

Some solutions for n=4 k=4
..0..0..1..2....0..0..1..1....0..1..1..2....0..1..1..1....0..0..0..1
..0..0..1..2....1..1..1..1....0..1..2..2....0..1..2..2....1..1..1..1
..1..1..1..2....1..2..2..2....1..1..2..2....0..1..2..2....1..2..2..2
..1..2..2..2....2..2..2..3....1..2..2..2....1..1..2..3....2..2..2..3

Crossrefs

Column 1 is A000125(n-1)

Sequence in context: A115216 A208637 A341867 * A229402 A266249 A276299

Adjacent sequences: A252935 A252936 A252937 * A252939 A252940 A252941

Keyword

nonn,tabl

Author

R. H. Hardin, Dec 24 2014

Status

approved