A253223 T(n,k) = number of n X k nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 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, 1, 0, 1, 3, 1, 1, 3, 6, 9, 1, 9, 6, 10, 25, 19, 19, 25, 10, 15, 49, 102, 19, 102, 49, 15, 21, 81, 263, 268, 268, 263, 81, 21, 28, 121, 504, 1249, 268, 1249, 504, 121, 28, 36, 169, 825, 3140, 3568, 3568, 3140, 825, 169, 36, 45, 225, 1226, 5986, 16028, 3568, 16028

Offset: 1

Author

R. H. Hardin, Dec 29 2014

Keywords

• nonn
• tabl

Examples

			Table starts:
..0...0....1.....3......6......10......15.......21........28........36
..0...0....1.....9.....25......49......81......121.......169.......225
..1...1....1....19....102.....263.....504......825......1226......1707
..3...9...19....19....268....1249....3140.....5986......9792.....14558
..6..25..102...268....268....3568...16028....40238.....77063....126673
.10..49..263..1249...3568....3568...47698...213155....538444...1039060
.15..81..504..3140..16028...47698...47698...649712...2913793...7415837
.21.121..825..5986..40238..213155..649712...649712...9023385..40680959
.28.169.1226..9792..77063..538444.2913793..9023385...9023385.127419681
.36.225.1707.14558.126673.1039060.7415837.40680959.127419681.127419681
Some solutions for n=4 and k=4:
  0 0 1 1   0 0 0 1   0 0 0 1   0 0 0 1   0 0 1 1
  0 0 1 1   1 1 1 1   0 0 0 1   0 0 0 1   0 0 1 1
  1 1 1 1   1 1 1 1   0 1 1 1   0 0 1 1   0 1 1 1
  1 1 1 1   1 1 1 1   1 1 1 1   1 1 1 1   1 1 1 1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..2812
• 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

Crossrefs

Column 1 is A000217(n-2).

Column 2 is A016754(n-3).

Formula

Empirical for column k:

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

k=2: a(n) = 4*n^2 - 20*n + 25 for n>2.

k=3: a(n) = 40*n^2 - 279*n + 497 for n>4.

k=4: a(n) = 480*n^2 - 4354*n + 10098 for n>6.

k=5: a(n) = 6400*n^2 - 71990*n + 206573 for n>8.

k=6: a(n) = 90112*n^2 - 1212288*n + 4150790 for n>10.

k=7: a(n) = 1306624*n^2 - 20460244*n + 81385043 for n>12.