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%I #6 Feb 14 2015 13:14:04
%S 9,26,27,66,75,81,147,168,216,243,294,331,441,622,729,540,597,789,
%T 1137,1791,2187,927,1008,1302,1905,2907,5157,6561,1507,1616,2032,2951,
%U 4429,7498,14849,19683,2343,2484,3042,4338,6582,10125,19338,42756,59049,3510
%N T(n,k)=Number of length n+k 0..2 arrays with at most one downstep in every k consecutive neighbor pairs
%C Table starts
%C ......9.....26.....66....147....294....540....927...1507...2343...3510...5096
%C .....27.....75....168....331....597...1008...1616...2484...3687...5313...7464
%C .....81....216....441....789...1302...2032...3042...4407...6215...8568..11583
%C ....243....622...1137...1905...2951...4338...6141...8448..11361..14997..19489
%C ....729...1791...2907...4429...6582...9297..12662..16779..21765..27753..34893
%C ...2187...5157...7498..10125..14001..19263..25578..33063..41851..52092..63954
%C ...6561..14849..19338..23463..29147..38010..49611..63075..78552..96210.116236
%C ..19683..42756..49698..55246..61542..73278..91887.115470.142200.172264.205869
%C ..59049.123111.127871.129480.133392.143045.166290.202716.247600.297597.352935
%C .177147.354484.329325.300432.292534.288057.303969.348070.415308.496188.585101
%H R. H. Hardin, <a href="/A255107/b255107.txt">Table of n, a(n) for n = 1..9999</a>
%F Empirical for column k:
%F k=1: a(n) = 3*a(n-1)
%F k=2: a(n) = 3*a(n-1) -a(n-3)
%F k=3: a(n) = 3*a(n-1) -3*a(n-2) +8*a(n-3) -9*a(n-4) +3*a(n-5) -a(n-6)
%F k=4: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +12*a(n-4) -18*a(n-5) +7*a(n-6) -3*a(n-8) +a(n-9)
%F k=5: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +18*a(n-5) -29*a(n-6) +12*a(n-7) -6*a(n-10) +3*a(n-11)
%F k=6: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +25*a(n-6) -42*a(n-7) +18*a(n-8) -10*a(n-12) +6*a(n-13)
%F k=7: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +33*a(n-7) -57*a(n-8) +25*a(n-9) -15*a(n-14) +10*a(n-15)
%F Empirical for row n:
%F n=1: a(n) = (1/120)*n^5 + (1/6)*n^4 + (19/24)*n^3 + (11/6)*n^2 + (16/5)*n + 3
%F n=2: a(n) = (1/120)*n^5 + (5/24)*n^4 + (37/24)*n^3 + (175/24)*n^2 + (239/20)*n + 6
%F n=3: a(n) = (1/120)*n^5 + (1/4)*n^4 + (59/24)*n^3 + (93/4)*n^2 + (1321/30)*n + 11
%F n=4: a(n) = (1/120)*n^5 + (7/24)*n^4 + (85/24)*n^3 + (1505/24)*n^2 + (2809/20)*n + 30 for n>2
%F n=5: a(n) = (1/120)*n^5 + (1/3)*n^4 + (115/24)*n^3 + (889/6)*n^2 + (3867/10)*n + 111 for n>3
%F n=6: a(n) = (1/120)*n^5 + (3/8)*n^4 + (149/24)*n^3 + (2521/8)*n^2 + (56417/60)*n + 385 for n>4
%F n=7: a(n) = (1/120)*n^5 + (5/12)*n^4 + (187/24)*n^3 + (7393/12)*n^2 + (20667/10)*n + 1143 for n>5
%e Some solutions for n=4 k=4
%e ..0....0....1....0....1....0....0....0....0....1....0....1....1....0....0....1
%e ..0....1....2....0....2....2....0....1....1....0....0....2....2....1....1....1
%e ..0....2....0....1....0....2....1....0....0....1....1....0....2....0....2....1
%e ..0....0....0....2....0....0....2....0....0....1....1....0....2....0....2....2
%e ..2....0....2....0....1....2....2....0....1....2....1....0....0....2....2....2
%e ..2....0....2....1....2....2....2....2....1....1....1....1....1....2....0....0
%e ..0....2....1....1....0....2....1....0....2....2....2....0....2....2....0....1
%e ..0....0....1....1....0....0....2....1....2....2....1....2....2....2....1....1
%Y Column 1 is A000244(n+1)
%Y Column 2 is A018919(n+1)
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Feb 14 2015
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