[seqfan] Re: A193376 Tabl = 20 existing sequences

[Ron Hardin]

Please add it to the series!

The same problem with 3X1 tiles apparently gives a n-1 n-3 recurrence (b-file 
still in progress), needs a formula too:

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

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

Column 1 is A000930
Column 2 is A003229(n-1)
Column 3 is A084386
Column 4 is A089977
Column 10 is A178205
Row 3 is A000027(n+1)
Row 4 is A004273(n+1)
Row 5 is A016777
Row 6 is A028872(n+2)
Row 7 is A144390(n+1)
Row 8 is A003154(n+1)


 rhhardin at mindspring.com
rhhardin at att.net (either)



----- Original Message ----
> From: "israel at math.ubc.ca" <israel at math.ubc.ca>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Mon, July 25, 2011 12:24:51 AM
> Subject: [seqfan] Re: A193376 Tabl = 20 existing sequences
> 
> This satisfies the recursion T(n+2,k) = T(n+1,k) + k T(n,k) with T(0,k) = 1,  
>T(1,k) = 1. Maple's solution is
> 
> T(n,k) =  ((2 k/(sqrt(1+4 k) -  1))^(n+1) - (-2 k/(sqrt(1+4 k) + 
>1))^(n+1))/sqrt(1+4 k)
> 
> Robert  Israel                                 israel at math.ubc.ca
> Department of  Mathematics        http://www.math.ubc.ca/~israel University of 
>British  Columbia            Vancouver, BC,  Canada