A193515 T(n,k) = number of ways to place any number of 3X1 tiles of k distinguishable colors into an nX1 grid.

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 7, 7, 6, 1, 1, 6, 9, 10, 13, 9, 1, 1, 7, 11, 13, 22, 23, 13, 1, 1, 8, 13, 16, 33, 43, 37, 19, 1, 1, 9, 15, 19, 46, 69, 73, 63, 28, 1, 1, 10, 17, 22, 61, 101, 121, 139, 109, 41, 1, 1, 11, 19, 25, 78, 139, 181, 253, 268, 183, 60, 1, 1, 12

Offset: 1

Author

R. H. Hardin, with proof and formula from Robert Israel in the Sequence Fans Mailing List, Jul 29 2011

Keywords

• nonn
• tabl

Comments

Table starts:

..1...1...1...1...1....1....1....1....1....1....1....1.....1.....1.....1.....1

..1...1...1...1...1....1....1....1....1....1....1....1.....1.....1.....1.....1

..2...3...4...5...6....7....8....9...10...11...12...13....14....15....16....17

..3...5...7...9..11...13...15...17...19...21...23...25....27....29....31....33

..4...7..10..13..16...19...22...25...28...31...34...37....40....43....46....49

..6..13..22..33..46...61...78...97..118..141..166..193...222...253...286...321

..9..23..43..69.101..139..183..233..289..351..419..493...573...659...751...849

.13..37..73.121.181..253..337..433..541..661..793..937..1093..1261..1441..1633

.19..63.139.253.411..619..883.1209.1603.2071.2619.3253..3979..4803..5731..6769

.28.109.268.529.916.1453.2164.3073.4204.5581.7228.9169.11428.14029.16996.20353

Examples

			Some solutions for n=7 k=3; colors=1,2,3 and empty=0
..3....0....0....2....0....1....3....0....0....0....1....0....3....1....0....0
..3....0....0....2....2....1....3....2....1....0....1....3....3....1....0....0
..3....1....0....2....2....1....3....2....1....2....1....3....3....1....0....3
..1....1....3....0....2....0....0....2....1....2....3....3....0....2....0....3
..1....1....3....0....0....2....2....2....1....2....3....2....1....2....1....3
..1....0....3....0....0....2....2....2....1....0....3....2....1....2....1....0
..0....0....0....0....0....2....2....2....1....0....0....2....1....0....1....0
		

Links

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

Crossrefs

Column 1 is A000930,

Column 2 is A003229(n-1),

Column 3 is A084386,

Column 4 is A089977,

Column 10 is A178205,

Row 6 is A028872(n+2),

Row 7 is A144390(n+1),

Row 8 is A003154(n+1).

Programs

Maple

T:= proc(n, k) option remember;
      `if`(n<0, 0,
      `if`(n<3 or k=0, 1, k*T(n-3, k) +T(n-1, k)))
    end:
seq(seq(T(n, d+1-n), n=1..d), d=1..13); # Alois P. Heinz, Jul 29 2011

Mathematica

nmax = 13; t[?Negative, ] = 0; t[n_, k_] /; (n < 3 || k == 0) = 1; t[n_, k_] := t[n, k] = k*t[n-3, k] + t[n-1, k]; Flatten[ Table[ t[n-k+1, k], {n , 1, nmax}, {k, n, 1, -1}]](* Jean-François Alcover, Nov 28 2011, after Maple *)

Formula

With z X 1 tiles of k colors on an n X 1 grid (with n >= z), either there is a tile (of any of the k colors) on the first spot, followed by any configuration on the remaining (n-z) X 1 grid, or the first spot is vacant, followed by any configuration on the remaining (n-1) X 1. So T(n,k) = T(n-1,k) + k*T(n-z,k), with T(n,k) = 1 for n=0,1,...,z-1. The solution is T(n,k) = sum_r r^(-n-1)/(1 + z k r^(z-1)) where the sum is over the roots of the polynomial k x^z + x - 1.

T(n,k) = sum {s=0..[n/3]} (binomial(n-2*s,s)*k^s).

For z X 1 tiles, T(n,k,z) = sum{s=0..[n/z]} (binomial(n-(z-1)*s,s)*k^s). - R. H. Hardin, Jul 31 2011