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A229445 T(n,k)=Number of nXk 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, and rows and columns lexicographically nondecreasing.

%I A229445 #6 Jul 23 2025 05:45:26
%S A229445 3,4,5,5,7,8,6,10,13,12,7,14,22,25,17,8,19,37,53,47,23,9,25,60,109,
%T A229445 128,84,30,10,32,93,212,324,293,142,38,11,40,138,387,753,915,625,228,
%U A229445 47,12,49,197,665,1609,2546,2402,1244,350,57,13,59,272,1083,3184,6374,8024
%N A229445 T(n,k)=Number of nXk 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, and rows and columns lexicographically nondecreasing.
%C A229445 Table starts
%C A229445 ..3...4....5....6.....7.....8......9.....10......11......12......13.......14
%C A229445 ..5...7...10...14....19....25.....32.....40......49......59......70.......82
%C A229445 ..8..13...22...37....60....93....138....197.....272.....365.....478......613
%C A229445 .12..25...53..109...212...387....665...1083....1684....2517....3637.....5105
%C A229445 .17..47..128..324...753..1609...3184...5890...10281...17075...27176....41696
%C A229445 .23..84..293..915..2546..6374..14536..30571...59969..110816..194535...326723
%C A229445 .30.142..625.2402..8024.23610..62205.149031..329106..677706.1314145..2419348
%C A229445 .38.228.1244.5843.23428.81177.247607.676983.1685570.3873314.8307126.16784531
%H A229445 R. H. Hardin, <a href="/A229445/b229445.txt">Table of n, a(n) for n = 1..480</a>
%F A229445 Empirical for column k:
%F A229445 k=1: a(n) = (1/2)*n^2 + (1/2)*n + 2
%F A229445 k=2: a(n) = (1/24)*n^4 + (1/12)*n^3 - (1/24)*n^2 + (23/12)*n + 2
%F A229445 k=3: [polynomial of degree 6]
%F A229445 k=4: [polynomial of degree 8]
%F A229445 k=5: [polynomial of degree 10]
%F A229445 k=6: [polynomial of degree 12]
%F A229445 k=7: [polynomial of degree 14]
%F A229445 Empirical for row n:
%F A229445 n=1: a(n) = n + 2
%F A229445 n=2: a(n) = (1/2)*n^2 + (1/2)*n + 4
%F A229445 n=3: a(n) = (1/3)*n^3 + (8/3)*n + 5
%F A229445 n=4: a(n) = (1/4)*n^4 - (1/3)*n^3 + (13/4)*n^2 + (11/6)*n + 7
%F A229445 n=5: a(n) = (11/60)*n^5 - (1/2)*n^4 + (15/4)*n^3 - n^2 + (257/30)*n + 6
%F A229445 n=6: [polynomial of degree 6]
%F A229445 n=7: [polynomial of degree 7]
%e A229445 Some solutions for n=4 k=4
%e A229445 ..0..2..2..2....0..2..2..2....0..0..2..2....0..0..2..2....0..2..2..2
%e A229445 ..1..0..0..2....1..0..0..0....0..0..2..2....1..1..0..0....0..2..2..2
%e A229445 ..2..1..1..0....2..1..1..1....1..1..0..0....1..1..1..1....0..2..2..2
%e A229445 ..2..1..1..1....2..2..2..2....1..1..1..1....2..2..1..1....1..0..0..2
%Y A229445 Column 1 is A022856(n+4)
%Y A229445 Row 2 is A145018(n+1)
%K A229445 nonn,tabl
%O A229445 1,1
%A A229445 _R. H. Hardin_ Sep 23 2013