A255107 T(n,k)=Number of length n+k 0..2 arrays with at most one downstep in every k consecutive neighbor pairs

9, 26, 27, 66, 75, 81, 147, 168, 216, 243, 294, 331, 441, 622, 729, 540, 597, 789, 1137, 1791, 2187, 927, 1008, 1302, 1905, 2907, 5157, 6561, 1507, 1616, 2032, 2951, 4429, 7498, 14849, 19683, 2343, 2484, 3042, 4338, 6582, 10125, 19338, 42756, 59049, 3510

Offset

1,1

Comments

Table starts

......9.....26.....66....147....294....540....927...1507...2343...3510...5096

.....27.....75....168....331....597...1008...1616...2484...3687...5313...7464

.....81....216....441....789...1302...2032...3042...4407...6215...8568..11583

....243....622...1137...1905...2951...4338...6141...8448..11361..14997..19489

....729...1791...2907...4429...6582...9297..12662..16779..21765..27753..34893

...2187...5157...7498..10125..14001..19263..25578..33063..41851..52092..63954

...6561..14849..19338..23463..29147..38010..49611..63075..78552..96210.116236

..19683..42756..49698..55246..61542..73278..91887.115470.142200.172264.205869

..59049.123111.127871.129480.133392.143045.166290.202716.247600.297597.352935

.177147.354484.329325.300432.292534.288057.303969.348070.415308.496188.585101

Links

R. H. Hardin, Table of n, a(n) for n = 1..9999

Formula

Empirical for column k:

k=1: a(n) = 3*a(n-1)

k=2: a(n) = 3*a(n-1) -a(n-3)

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)

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)

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)

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)

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)

Empirical for row n:

n=1: a(n) = (1/120)*n^5 + (1/6)*n^4 + (19/24)*n^3 + (11/6)*n^2 + (16/5)*n + 3

n=2: a(n) = (1/120)*n^5 + (5/24)*n^4 + (37/24)*n^3 + (175/24)*n^2 + (239/20)*n + 6

n=3: a(n) = (1/120)*n^5 + (1/4)*n^4 + (59/24)*n^3 + (93/4)*n^2 + (1321/30)*n + 11

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

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

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

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

Example

Some solutions for n=4 k=4
..0....0....1....0....1....0....0....0....0....1....0....1....1....0....0....1
..0....1....2....0....2....2....0....1....1....0....0....2....2....1....1....1
..0....2....0....1....0....2....1....0....0....1....1....0....2....0....2....1
..0....0....0....2....0....0....2....0....0....1....1....0....2....0....2....2
..2....0....2....0....1....2....2....0....1....2....1....0....0....2....2....2
..2....0....2....1....2....2....2....2....1....1....1....1....1....2....0....0
..0....2....1....1....0....2....1....0....2....2....2....0....2....2....0....1
..0....0....1....1....0....0....2....1....2....2....1....2....2....2....1....1

Crossrefs

Column 1 is A000244(n+1)

Column 2 is A018919(n+1)

Sequence in context: A144114 A209969 A144701 * A022421 A075395 A352775

Adjacent sequences: A255104 A255105 A255106 * A255108 A255109 A255110

Keyword

nonn,tabl

Author

R. H. Hardin, Feb 14 2015

Status

approved