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A212829 T(n,k)=Number of 0..k arrays of length n with no adjacent pair equal to its immediately preceding adjacent pair, and new values introduced in 0..k order.

%I A212829 #6 Jul 22 2025 22:58:16
%S A212829 1,1,2,1,2,4,1,2,5,6,1,2,5,12,10,1,2,5,13,33,16,1,2,5,13,43,90,26,1,2,
%T A212829 5,13,44,152,246,42,1,2,5,13,44,167,559,672,68,1,2,5,13,44,168,695,
%U A212829 2091,1836,110,1,2,5,13,44,168,716,3070,7882,5016,178,1,2,5,13,44,168,717
%N A212829 T(n,k)=Number of 0..k arrays of length n with no adjacent pair equal to its immediately preceding adjacent pair, and new values introduced in 0..k order.
%C A212829 Table starts
%C A212829 ...1.....1......1.......1.......1.......1.......1.......1.......1.......1
%C A212829 ...2.....2......2.......2.......2.......2.......2.......2.......2.......2
%C A212829 ...4.....5......5.......5.......5.......5.......5.......5.......5.......5
%C A212829 ...6....12.....13......13......13......13......13......13......13......13
%C A212829 ..10....33.....43......44......44......44......44......44......44......44
%C A212829 ..16....90....152.....167.....168.....168.....168.....168.....168.....168
%C A212829 ..26...246....559.....695.....716.....717.....717.....717.....717.....717
%C A212829 ..42...672...2091....3070....3331....3359....3360....3360....3360....3360
%C A212829 ..68..1836...7882...14074...16599...17055...17091...17092...17092...17092
%C A212829 .110..5016..29809...65958...87059...92749...93492...93537...93538...93538
%C A212829 .178.13704.112895..313098..473569..534071..545670..546817..546872..546873
%C A212829 .288.37440.427824.1497216.2641428.3220152.3372738.3394616.3396312.3396378
%H A212829 R. H. Hardin, <a href="/A212829/b212829.txt">Table of n, a(n) for n = 1..6054</a>
%F A212829 Empirical for column k:
%F A212829 k=1: a(n) = a(n-1) +a(n-2) for n>3
%F A212829 k=2: a(n) = 2*a(n-1) +2*a(n-2) for n>5
%F A212829 k=3: a(n) = 4*a(n-1) +a(n-2) -6*a(n-3) -3*a(n-4) for n>7
%F A212829 k=4: a(n) = 7*a(n-1) -7*a(n-2) -20*a(n-3) +10*a(n-4) +24*a(n-5) +8*a(n-6) for n>9
%F A212829 k=5: a(n) = 11*a(n-1) -30*a(n-2) -21*a(n-3) +112*a(n-4) +63*a(n-5) -119*a(n-6) -120*a(n-7) -30*a(n-8) for n>11
%e A212829 Some solutions for n=8 k=4
%e A212829 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A212829 ..0....1....1....1....1....1....0....1....1....1....0....1....1....1....1....1
%e A212829 ..1....2....2....2....2....1....1....2....2....2....0....2....2....0....2....0
%e A212829 ..0....3....3....2....1....0....2....3....1....0....1....3....1....2....3....2
%e A212829 ..0....4....3....1....3....2....3....1....0....1....2....0....0....1....4....2
%e A212829 ..1....3....2....1....4....0....1....3....3....3....0....2....0....0....0....2
%e A212829 ..2....3....1....3....2....1....0....2....3....4....1....2....3....0....0....3
%e A212829 ..1....1....4....0....0....2....3....4....1....1....0....2....4....3....4....3
%Y A212829 Column 1 is A128588
%K A212829 nonn,tabl
%O A212829 1,3
%A A212829 _R. H. Hardin_ May 28 2012