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

Author

R. H. Hardin, Feb 14 2015

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

Crossrefs

Column 1 is A000244(n+1)

Column 2 is A018919(n+1)

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