login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A193517 T(n,k) = number of ways to place any number of 5X1 tiles of k distinguishable colors into an nX1 grid. 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 5, 7, 7, 5, 1, 1, 1, 1, 6, 9, 10, 9, 6, 1, 1, 1, 1, 7, 11, 13, 13, 11, 8, 1, 1, 1, 1, 8, 13, 16, 17, 16, 17, 11, 1, 1, 1, 1, 9, 15, 19, 21, 21, 28, 27, 15, 1, 1, 1, 1, 10, 17, 22, 25, 26, 41, 49, 41, 20, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,15

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

..2..3...4...5...6...7...8....9...10...11...12...13...14...15...16...17...18

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

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

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

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

..8.17..28..41..56..73..92..113..136..161..188..217..248..281..316..353..392

.11.27..49..77.111.151.197..249..307..371..441..517..599..687..781..881..987

.15.41..79.129.191.265.351..449..559..681..815..961.1119.1289.1471.1665.1871

.20.59.118.197.296.415.554..713..892.1091.1310.1549.1808.2087.2386.2705.3044

.26.81.166.281.426.601.806.1041.1306.1601.1926.2281.2666.3081.3526.4001.4506

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/5]} (binomial(n-4*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=11 k=3; colors=1, 2, 3; empty=0

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

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

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

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

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

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

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

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

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

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

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

MAPLE

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

      `if`(n<0, 0,

      `if`(n<5 or k=0, 1, k*T(n-5, 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 < 5 || k == 0, 1, k*T[n-5, 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 A003520,

Column 2 is A143447(n-4),

Column 3 is A143455(n-4),

Row 10 is A028884.

Sequence in context: A215625 A260222 A181386 * A189006 A245013 A219924

Adjacent sequences:  A193514 A193515 A193516 * A193518 A193519 A193520

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 18 14:15 EST 2017. Contains 294893 sequences.