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A243641 T(n,k)=Number of length n+2 0..k arrays with no three unequal elements in a row and new values 0..k introduced in 0..k order.

%I A243641 #6 Jul 23 2025 11:21:37
%S A243641 4,4,8,4,9,16,4,9,21,32,4,9,21,50,64,4,9,21,51,120,128,4,9,21,51,127,
%T A243641 289,256,4,9,21,51,127,323,697,512,4,9,21,51,127,324,835,1682,1024,4,
%U A243641 9,21,51,127,324,844,2187,4060,2048,4,9,21,51,127,324,844,2242,5787,9801,4096,4
%N A243641 T(n,k)=Number of length n+2 0..k arrays with no three unequal elements in a row and new values 0..k introduced in 0..k order.
%C A243641 Table starts
%C A243641 ....4....4.....4.....4.....4.....4.....4.....4.....4.....4.....4.....4.....4
%C A243641 ....8....9.....9.....9.....9.....9.....9.....9.....9.....9.....9.....9.....9
%C A243641 ...16...21....21....21....21....21....21....21....21....21....21....21....21
%C A243641 ...32...50....51....51....51....51....51....51....51....51....51....51....51
%C A243641 ...64..120...127...127...127...127...127...127...127...127...127...127...127
%C A243641 ..128..289...323...324...324...324...324...324...324...324...324...324...324
%C A243641 ..256..697...835...844...844...844...844...844...844...844...844...844...844
%C A243641 ..512.1682..2187..2242..2243..2243..2243..2243..2243..2243..2243..2243..2243
%C A243641 .1024.4060..5787..6062..6073..6073..6073..6073..6073..6073..6073..6073..6073
%C A243641 .2048.9801.15435.16655.16736.16737.16737.16737.16737.16737.16737.16737.16737
%H A243641 R. H. Hardin, <a href="/A243641/b243641.txt">Table of n, a(n) for n = 1..9999</a>
%F A243641 Empirical for column k:
%F A243641 k=1: a(n) = 2*a(n-1)
%F A243641 k=2: a(n) = 3*a(n-1) -a(n-2) -a(n-3)
%F A243641 k=3: a(n) = 5*a(n-1) -6*a(n-2) -2*a(n-3) +4*a(n-4)
%F A243641 k=4: a(n) = 7*a(n-1) -14*a(n-2) +21*a(n-4) -7*a(n-5) -6*a(n-6)
%F A243641 k=5: [order 8]
%F A243641 k=6: [order 10]
%F A243641 k=7: [order 12]
%F A243641 k=8: [order 14]
%F A243641 k=9: [order 16]
%e A243641 Some solutions for n=6 k=4
%e A243641 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A243641 ..1....0....0....0....1....1....0....0....0....0....1....1....0....1....1....0
%e A243641 ..1....1....1....0....1....0....0....0....1....1....1....0....0....1....1....1
%e A243641 ..1....1....0....1....1....0....1....0....0....1....2....0....1....2....2....1
%e A243641 ..2....2....0....0....0....0....0....0....0....2....1....1....1....2....2....0
%e A243641 ..2....1....1....1....1....1....0....0....0....1....1....1....0....1....3....1
%e A243641 ..3....1....1....1....0....0....0....0....1....1....1....1....0....1....3....1
%e A243641 ..2....2....1....1....1....1....0....1....1....1....0....0....2....2....0....0
%Y A243641 Column 1 is A000079(n+1)
%Y A243641 Column 2 is A024537(n+1)
%Y A243641 Column 3 is A094286(n+2)
%K A243641 nonn,tabl
%O A243641 1,1
%A A243641 _R. H. Hardin_, Jun 08 2014