A243519 T(n,k)=Number of length n+2 0..k arrays with no three elements in a row with pattern aba or abb (with a!=b) and new values 0..k introduced in 0..k order.

2, 3, 2, 3, 4, 2, 3, 5, 5, 2, 3, 5, 9, 6, 2, 3, 5, 10, 17, 7, 2, 3, 5, 10, 24, 33, 8, 2, 3, 5, 10, 25, 65, 65, 9, 2, 3, 5, 10, 25, 76, 187, 129, 10, 2, 3, 5, 10, 25, 77, 263, 552, 257, 11, 2, 3, 5, 10, 25, 77, 279, 978, 1646, 513, 12, 2, 3, 5, 10, 25, 77, 280, 1134, 3773, 4927, 1025, 13

Offset: 1

Author

R. H. Hardin, Jun 05 2014

Keywords

• nonn
• tabl

Comments

Table starts

.2..3....3.....3.....3......3......3......3......3......3......3......3......3

.2..4....5.....5.....5......5......5......5......5......5......5......5......5

.2..5....9....10....10.....10.....10.....10.....10.....10.....10.....10.....10

.2..6...17....24....25.....25.....25.....25.....25.....25.....25.....25.....25

.2..7...33....65....76.....77.....77.....77.....77.....77.....77.....77.....77

.2..8...65...187...263....279....280....280....280....280....280....280....280

.2..9..129...552...978...1134...1156...1157...1157...1157...1157...1157...1157

.2.10..257..1646..3773...4979...5267...5296...5297...5297...5297...5297...5297

.2.11..513..4927.14824..22981..25915..26406..26443..26444..26444..26444..26444

.2.12.1025.14769.58771.109453.135214.141585.142372.142418.142419.142419.142419

Examples

			All solutions for n=3 k=4
..0....0....0....0....0....0....0....0....0....0
..1....1....0....1....0....0....1....0....0....1
..2....2....0....2....1....1....2....0....0....2
..3....0....0....3....2....2....3....1....0....0
..4....1....0....0....0....3....1....2....1....3
		

Links

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

Crossrefs

Column 3 is A000051

Diagonal is A005001(n+1)

Formula

Empirical for column k:

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

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

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

k=4: a(n) = 5*a(n-1) -7*a(n-2) +3*a(n-3)

k=5: a(n) = 8*a(n-1) -21*a(n-2) +22*a(n-3) -8*a(n-4)

k=6: a(n) = 12*a(n-1) -52*a(n-2) +102*a(n-3) -91*a(n-4) +30*a(n-5)

k=7: a(n) = 17*a(n-1) -111*a(n-2) +355*a(n-3) -584*a(n-4) +468*a(n-5) -144*a(n-6)

k=8: a(n) = 23*a(n-1) -212*a(n-2) +1010*a(n-3) -2669*a(n-4) +3887*a(n-5) -2878*a(n-6) +840*a(n-7)

k=9: a(n) = 30*a(n-1) -372*a(n-2) +2478*a(n-3) -9639*a(n-4) +22260*a(n-5) -29588*a(n-6) +20592*a(n-7) -5760*a(n-8)