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A214101 T(n,k)=Number of 0..2 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..2 introduced in row major order.

%I A214101 #6 Jul 22 2025 23:07:11
%S A214101 1,1,3,3,2,9,5,19,4,27,11,30,121,8,81,21,143,180,771,16,243,43,322,
%T A214101 2041,1080,4913,32,729,85,1179,5068,29540,6480,31307,64,2187,171,3110,
%U A214101 37441,79968,428383,38880,199497,128,6561,341,10183,121588,1241355,1262128
%N A214101 T(n,k)=Number of 0..2 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..2 introduced in row major order.
%C A214101 Table starts
%C A214101 ..1..1....3....5.....11......21.......43........85........171.........341
%C A214101 ..3..2...19...30....143.....322.....1179......3110......10183.......28842
%C A214101 ..9..4..121..180...2041....5068....37441....121588.....722009.....2720828
%C A214101 .27..8..771.1080..29540...79968..1241355...4807928...54733587...263068168
%C A214101 .81.16.4913.6480.428383.1262128.41634729.190532944.4254090231.25595530224
%H A214101 R. H. Hardin, <a href="/A214101/b214101.txt">Table of n, a(n) for n = 1..309</a>
%F A214101 Empirical for column k:
%F A214101 k=1: a(n) = 3*a(n-1)
%F A214101 k=2: a(n) = 2*a(n-1)
%F A214101 k=3: a(n) = 7*a(n-1) -4*a(n-2)
%F A214101 k=4: a(n) = 6*a(n-1)
%F A214101 k=5: a(n) = 19*a(n-1) -71*a(n-2) +86*a(n-3) -24*a(n-4)
%F A214101 k=6: a(n) = 18*a(n-1) -36*a(n-2) +16*a(n-3)
%F A214101 k=7: a(n) = 54*a(n-1) -820*a(n-2) +4906*a(n-3) -11803*a(n-4) +11888*a(n-5) -4672*a(n-6) +576*a(n-7)
%F A214101 Empirical for row n:
%F A214101 n=1: a(k)=a(k-1)+2*a(k-2)
%F A214101 n=2: a(k)=2*a(k-1)+5*a(k-2)-6*a(k-3)
%F A214101 n=3: a(k)=3*a(k-1)+15*a(k-2)-33*a(k-3)-22*a(k-4)+38*a(k-5)+8*a(k-6)-8*a(k-7)
%F A214101 n=4: (order 11)
%F A214101 n=5: (order 29)
%F A214101 n=6: (order 40)
%e A214101 Some solutions for n=4 k=1
%e A214101 ..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
%e A214101 ..2..0....2..0....1..0....1..2....1..2....1..2....1..2....2..0....1..0....1..2
%e A214101 ..0..1....1..2....0..1....0..1....2..0....2..0....2..0....0..2....2..1....0..1
%e A214101 ..1..2....2..0....2..0....2..0....0..2....1..2....0..1....1..0....1..2....1..0
%Y A214101 Column 3 is A138977
%Y A214101 Column 4 is A052934
%Y A214101 Row 1 is A001045
%Y A214101 Row 2 is A094554(n+1)
%K A214101 nonn,tabl
%O A214101 1,3
%A A214101 _R. H. Hardin_ Jul 04 2012