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A193518 T(n,k) = number of ways to place any number of 6X1 tiles of k distinguishable colors into an nX1 grid. 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 1, 5, 7, 7, 5, 1, 1, 1, 1, 1, 6, 9, 10, 9, 6, 1, 1, 1, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 1, 1, 1, 8, 13, 16, 17, 16, 13, 9, 1, 1, 1, 1, 1, 9, 15, 19, 21, 21, 19, 19, 12, 1, 1, 1, 1, 1, 10, 17 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,21

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....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....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...18...19

..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37

..4..7..10..13..16..19..22..25..28...31...34...37...40...43...46...49...52...55

..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73

..6.11..16..21..26..31..36..41..46...51...56...61...66...71...76...81...86...91

..7.13..19..25..31..37..43..49..55...61...67...73...79...85...91...97..103..109

..9.19..31..45..61..79..99.121.145..171..199..229..261..295..331..369..409..451

.12.29..52..81.116.157.204.257.316..381..452..529..612..701..796..897.1004.1117

.16.43..82.133.196.271.358.457.568..691..826..973.1132.1303.1486.1681.1888.2107

.21.61.121.201.301.421.561.721.901.1101.1321.1561.1821.2101.2401.2721.3061.3421

LINKS

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

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/6]} (binomial(n-5*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

EXAMPLE

Some solutions for n=13 k=3; colors=1, 2, 3; empty=0

..0....0....0....0....0....3....0....0....0....0....2....0....0....2....2....1

..3....0....1....2....1....3....0....0....0....2....2....0....0....2....2....1

..3....0....1....2....1....3....0....0....0....2....2....0....0....2....2....1

..3....0....1....2....1....3....0....0....0....2....2....2....0....2....2....1

..3....0....1....2....1....3....0....0....3....2....2....2....0....2....2....1

..3....1....1....2....1....3....0....0....3....2....2....2....0....2....2....1

..3....1....1....2....1....0....3....0....3....2....3....2....0....0....0....2

..1....1....1....0....0....1....3....3....3....3....3....2....2....2....0....2

..1....1....1....0....0....1....3....3....3....3....3....2....2....2....0....2

..1....1....1....0....0....1....3....3....3....3....3....0....2....2....0....2

..1....1....1....0....0....1....3....3....0....3....3....0....2....2....0....2

..1....0....1....0....0....1....3....3....0....3....3....0....2....2....0....2

..1....0....1....0....0....1....0....3....0....3....0....0....2....2....0....0

MAPLE

T:= proc(n, k) option remember;

      `if`(n<0, 0,

      `if`(n<6 or k=0, 1, k*T(n-6, 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

T[n_, k_] := T[n, k] = If[n<0, 0, If[n < 6 || k == 0, 1, k*T[n-6, k]+T[n-1, k]]]; Table[Table[T[n, d+1-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-Fran├žois Alcover, Mar 04 2014, after Alois P. Heinz *)

CROSSREFS

Column 1 is A005708,

Column 2 is A143448(n-5),

Column 3 is A143456(n-5),

Row 12 is A190576(n+1),

Row 15 is A069133(n+1).

Sequence in context: A253642 A070084 A268372 * A060176 A218220 A247487

Adjacent sequences:  A193515 A193516 A193517 * A193519 A193520 A193521

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified September 26 00:10 EDT 2017. Contains 292500 sequences.