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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.

%I A253223 #19 Apr 25 2024 07:06:57
%S A253223 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,
%T A253223 15,21,81,263,268,268,263,81,21,28,121,504,1249,268,1249,504,121,28,
%U A253223 36,169,825,3140,3568,3568,3140,825,169,36,45,225,1226,5986,16028,3568,16028
%N 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.
%H A253223 R. H. Hardin, <a href="/A253223/b253223.txt">Table of n, a(n) for n = 1..2812</a>
%H A253223 Robert Dougherty-Bliss, <a href="https://sites.math.rutgers.edu/~zeilberg/Theses/RobertDoughertyBlissThesis.pdf">Experimental Methods in Number Theory and Combinatorics</a>, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 21.
%H A253223 Robert Dougherty-Bliss and Manuel Kauers, <a href="https://arxiv.org/abs/2309.00487">Hardinian Arrays</a>, arXiv:2309.00487 [math.CO], 2023.<a href="https://doi.org/10.37236/12358">Hardinian Arrays</a>, El. J. Combinat. 31 (2) (2024) #P2.9
%F A253223 Empirical for column k:
%F A253223 k=1: a(n) = (1/2)*n^2 - (3/2)*n + 1.
%F A253223 k=2: a(n) = 4*n^2 - 20*n + 25 for n>2.
%F A253223 k=3: a(n) = 40*n^2 - 279*n + 497 for n>4.
%F A253223 k=4: a(n) = 480*n^2 - 4354*n + 10098 for n>6.
%F A253223 k=5: a(n) = 6400*n^2 - 71990*n + 206573 for n>8.
%F A253223 k=6: a(n) = 90112*n^2 - 1212288*n + 4150790 for n>10.
%F A253223 k=7: a(n) = 1306624*n^2 - 20460244*n + 81385043 for n>12.
%e A253223 Table starts:
%e A253223 ..0...0....1.....3......6......10......15.......21........28........36
%e A253223 ..0...0....1.....9.....25......49......81......121.......169.......225
%e A253223 ..1...1....1....19....102.....263.....504......825......1226......1707
%e A253223 ..3...9...19....19....268....1249....3140.....5986......9792.....14558
%e A253223 ..6..25..102...268....268....3568...16028....40238.....77063....126673
%e A253223 .10..49..263..1249...3568....3568...47698...213155....538444...1039060
%e A253223 .15..81..504..3140..16028...47698...47698...649712...2913793...7415837
%e A253223 .21.121..825..5986..40238..213155..649712...649712...9023385..40680959
%e A253223 .28.169.1226..9792..77063..538444.2913793..9023385...9023385.127419681
%e A253223 .36.225.1707.14558.126673.1039060.7415837.40680959.127419681.127419681
%e A253223 Some solutions for n=4 and k=4:
%e A253223   0 0 1 1   0 0 0 1   0 0 0 1   0 0 0 1   0 0 1 1
%e A253223   0 0 1 1   1 1 1 1   0 0 0 1   0 0 0 1   0 0 1 1
%e A253223   1 1 1 1   1 1 1 1   0 1 1 1   0 0 1 1   0 1 1 1
%e A253223   1 1 1 1   1 1 1 1   1 1 1 1   1 1 1 1   1 1 1 1
%Y A253223 Column 1 is A000217(n-2).
%Y A253223 Column 2 is A016754(n-3).
%K A253223 nonn,tabl
%O A253223 1,7
%A A253223 _R. H. Hardin_, Dec 29 2014