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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.

%I A252938 #6 Jul 23 2025 13:46:57
%S A252938 1,2,2,4,5,4,8,13,13,8,15,34,44,34,15,26,83,153,153,83,26,42,176,494,
%T A252938 711,494,176,42,64,329,1343,3067,3067,1343,329,64,93,558,3016,10920,
%U A252938 17962,10920,3016,558,93,130,879,5833,30818,86488,86488,30818,5833,879,130,176
%N 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.
%C A252938 Table starts
%C A252938 ...1....2.....4......8......15.......26.........42..........64...........93
%C A252938 ...2....5....13.....34......83......176........329.........558..........879
%C A252938 ...4...13....44....153.....494.....1343.......3016........5833........10114
%C A252938 ...8...34...153....711....3067....10920......30818.......70640.......138558
%C A252938 ..15...83...494...3067...17962....86488.....320270......917811......2127013
%C A252938 ..26..176..1343..10920...86488...578342....2952734....11219797.....32649081
%C A252938 ..42..329..3016..30818..320270..2952734...21312696...113154831....440052087
%C A252938 ..64..558..5833..70640..917811.11219797..113154831...857248091...4687944300
%C A252938 ..93..879.10114.138558.2127013.32649081..440052087..4687944300..36723156004
%C A252938 .130.1308.16179.242764.4211511.76641323.1302939451.18615501830.205553855458
%H A252938 R. H. Hardin, <a href="/A252938/b252938.txt">Table of n, a(n) for n = 1..1200</a>
%F A252938 Empirical for column k:
%F A252938 k=1: a(n) = (1/6)*n^3 - (1/2)*n^2 + (4/3)*n
%F A252938 k=2: a(n) = (8/3)*n^3 - 18*n^2 + (145/3)*n - 42 for n>2
%F A252938 k=3: a(n) = (160/3)*n^3 - 548*n^2 + (6071/3)*n - 2591 for n>4
%F A252938 k=4: a(n) = (4096/3)*n^3 - 18720*n^2 + (269642/3)*n - 149376 for n>6
%F A252938 k=5: a(n) = (133120/3)*n^3 - 760496*n^2 + (13526246/3)*n - 9199709 for n>8
%F A252938 k=6: [polynomial of degree 3] for n>10
%F A252938 k=7: [polynomial of degree 3] for n>12
%e A252938 Some solutions for n=4 k=4
%e A252938 ..0..0..1..2....0..0..1..1....0..1..1..2....0..1..1..1....0..0..0..1
%e A252938 ..0..0..1..2....1..1..1..1....0..1..2..2....0..1..2..2....1..1..1..1
%e A252938 ..1..1..1..2....1..2..2..2....1..1..2..2....0..1..2..2....1..2..2..2
%e A252938 ..1..2..2..2....2..2..2..3....1..2..2..2....1..1..2..3....2..2..2..3
%Y A252938 Column 1 is A000125(n-1)
%K A252938 nonn,tabl
%O A252938 1,2
%A A252938 _R. H. Hardin_, Dec 24 2014