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A252983 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value increasing by 0 or 1 with every step right, diagonally se or down.

%I A252983 #6 Jul 23 2025 13:50:05
%S A252983 0,0,0,1,0,1,3,1,1,3,6,13,1,13,6,10,41,33,33,41,10,15,85,266,68,266,
%T A252983 85,15,21,145,851,1247,1247,851,145,21,28,221,1836,8487,4657,8487,
%U A252983 1836,221,28,36,313,3221,27905,67537,67537,27905,3221,313,36,45,421,5006,62977
%N A252983 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value increasing by 0 or 1 with every step right, diagonally se or down.
%C A252983 Table starts
%C A252983 ..0...0....1......3.......6........10..........15............21.............28
%C A252983 ..0...0....1.....13......41........85.........145...........221............313
%C A252983 ..1...1....1.....33.....266.......851........1836..........3221...........5006
%C A252983 ..3..13...33.....68....1247......8487.......27905.........62977.........114433
%C A252983 ..6..41..266...1247....4657.....67537......433401.......1481460........3510600
%C A252983 .10..85..851...8487...67537....432842.....5672484......36112108......129234988
%C A252983 .15.145.1836..27905..433401...5672484....60650883.....766674140.....4970634131
%C A252983 .21.221.3221..62977.1481460..36112108...766674140...13458882036...170090480091
%C A252983 .28.313.5006.114433.3510600.129234988..4970634131..170090480091..4857082197177
%C A252983 .36.421.7191.182273.6637020.322183180.18692194423.1139074556531.62656851440792
%H A252983 R. H. Hardin, <a href="/A252983/b252983.txt">Table of n, a(n) for n = 1..480</a>
%F A252983 Empirical for column k:
%F A252983 k=1: a(n) = (1/2)*n^2 - (3/2)*n + 1
%F A252983 k=2: a(n) = 8*n^2 - 44*n + 61 for n>2
%F A252983 k=3: a(n) = 200*n^2 - 1615*n + 3341 for n>4
%F A252983 k=4: a(n) = 8192*n^2 - 87808*n + 241153 for n>6
%F A252983 k=5: a(n) = 557568*n^2 - 7467372*n + 25553940 for n>8
%F A252983 k=6: a(n) = 63438848*n^2 - 1019729920*n + 4176308004 for n>10
%F A252983 k=7: a(n) = 12103190528*n^2 - 226960984822*n + 1081747760523 for n>12
%e A252983 Some solutions for n=4 k=4
%e A252983 ..0..0..1..1....0..0..0..0....0..0..0..1....0..0..0..0....0..0..1..1
%e A252983 ..0..1..1..1....0..0..0..0....0..0..0..1....0..0..0..1....0..1..1..1
%e A252983 ..0..1..1..1....0..0..1..1....0..0..0..1....0..0..1..1....1..1..1..1
%e A252983 ..0..1..1..1....0..1..1..1....0..0..1..1....0..0..1..1....1..1..1..1
%Y A252983 Column 1 is A000217(n-2)
%Y A252983 Column 2 is A102083(n-3)
%K A252983 nonn,tabl
%O A252983 1,7
%A A252983 _R. H. Hardin_, Dec 25 2014