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A214112 T(n,k)=Number of 0..3 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..3 introduced in row major order.

%I A214112 #6 Jul 22 2025 23:08:30
%S A214112 1,1,4,4,11,25,10,111,121,172,31,670,3502,1331,1201,91,4994,44900,
%T A214112 110985,14641,8404,274,34041,825105,3008980,3517864,161051,58825,820,
%U A214112 241021,12777541,136579852,201647240,111505491,1771561,411772,2461,1678940
%N A214112 T(n,k)=Number of 0..3 colorings of an nx(k+1) array circular in the k+1 direction with new values 0..3 introduced in row major order.
%C A214112 Table starts
%C A214112 ....1.....1.......4........10..........31............91.............274
%C A214112 ....4....11.....111.......670........4994.........34041..........241021
%C A214112 ...25...121....3502.....44900......825105......12777541.......214404272
%C A214112 ..172..1331..110985...3008980...136579852....4797577911....191154162535
%C A214112 .1201.14641.3517864.201647240.22615881851.1801391900581.170522196557894
%H A214112 R. H. Hardin, <a href="/A214112/b214112.txt">Table of n, a(n) for n = 1..160</a>
%F A214112 Empirical for column k:
%F A214112 k=1: a(n) = 8*a(n-1) -7*a(n-2)
%F A214112 k=2: a(n) = 11*a(n-1)
%F A214112 k=3: a(n) = 35*a(n-1) -107*a(n-2) +73*a(n-3)
%F A214112 k=4: a(n) = 68*a(n-1) -66*a(n-2)
%F A214112 k=5: a(n) = 200*a(n-1) -5769*a(n-2) +11744*a(n-3) +43057*a(n-4) -89856*a(n-5) +40625*a(n-6)
%F A214112 k=6: a(n) = 416*a(n-1) -15454*a(n-2) +89758*a(n-3) +90848*a(n-4) -438718*a(n-5) +62801*a(n-6)
%F A214112 k=7: (order 15)
%F A214112 Empirical for row n:
%F A214112 n=1: a(k)=3*a(k-1)+a(k-2)-3*a(k-3)
%F A214112 n=2: a(k)=4*a(k-1)+22*a(k-2)-4*a(k-3)-21*a(k-4)
%F A214112 n=3: a(k)=11*a(k-1)+123*a(k-2)-509*a(k-3)-1615*a(k-4)+7137*a(k-5)-19*a(k-6)-20571*a(k-7)+13176*a(k-8)+13932*a(k-9)-11664*a(k-10)
%F A214112 n=4: (order 26)
%F A214112 n=5: (order 71)
%e A214112 Some solutions for n=4 k=1
%e A214112 ..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
%e A214112 ..1..0....1..2....2..3....1..0....1..0....1..0....1..2....1..0....1..0....1..2
%e A214112 ..0..1....0..1....3..1....0..1....2..3....2..1....3..0....0..2....2..3....3..1
%e A214112 ..1..2....1..0....1..0....1..0....3..2....3..0....0..1....1..3....3..1....0..2
%Y A214112 Column 1 is A034494(n-1)
%Y A214112 Column 2 is A001020(n-1)
%Y A214112 Row 1 is A006342(n-1)
%K A214112 nonn,tabl
%O A214112 1,3
%A A214112 _R. H. Hardin_ Jul 04 2012