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

1,7

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

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.

Example

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

Crossrefs

Column 1 is A000217(n-2).

Column 2 is A016754(n-3).

Sequence in context: A365968 A156710 A114588 * A121745 A252983 A089312

Adjacent sequences: A253220 A253221 A253222 * A253224 A253225 A253226

Keyword

nonn,tabl

Author

R. H. Hardin, Dec 29 2014

Status

approved