A272474 -id:A272474 - cp's OEIS frontend

A068924 Number of ways to tile a 5 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

6, 3, 2, 2, 4, 4, 6, 8, 10, 14, 18, 24, 32, 42, 56, 74, 98, 130, 172, 228, 302, 400, 530, 702, 930, 1232, 1632, 2162, 2864, 3794, 5026, 6658, 8820, 11684, 15478, 20504, 27162, 35982, 47666, 63144, 83648, 110810, 146792, 194458, 257602, 341250

Offset: 1

Dean Hickerson, Mar 11 2002

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 4.
F. Ruskey and J. Woodcock, Counting Fixed-Height Tatami Tilings, Electronic Journal of Combinatorics, Paper R126 (2009) 20 pages.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

Cf. A068930 for incongruent tilings, A068920 for more info. First column of A272474.

For n >= 6, a(n) = a(n-2) + a(n-3).
G.f.: x*(-6+x^4+7*x^3+4*x^2-3*x)/(-1+x^3+x^2). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by R. J. Mathar, Sep 16 2009]
a(n) = 2*A000931(n+3) for n>=3. - R. J. Mathar, Dec 06 2013

A192091 Number of tatami tilings of a 5 X n grid (with monomers allowed).

Original entry on oeis.org

1, 8, 68, 90, 126, 178, 325, 584, 1165, 2030, 3619, 6080, 10987, 19362, 35477, 62360, 111837, 195614, 350707, 619568, 1112315, 1967090, 3514597, 6214984, 11093549, 19664558, 35090115, 62247552, 110934699, 196859394, 350650261

Offset: 0

Frank Ruskey and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jul 07 2011

nonn

A tatami tiling consists of dimers (1 X 2) and monomers (1 X 1) where no four meet at a point.

			Here are some tatami tilings of the 5 X 3 grid:
    _ _ _ _ _   _ _ _ _ _   _ _ _ _ _   _ _ _ _ _
   |_ _| |_| | |_| |_|_ _| | |_ _| |_| |_| | |_ _|
   |_ _|_| |_| | |_|_ _| | |_| |_|_| | | |_|_| | |
   |_|_ _|_|_| |_|_ _|_|_| |_|_|_ _|_| |_|_ _|_|_|

A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, Monomer-Dimer Tatami Tilings of Rectangular Regions, Electronic Journal of Combinatorics, 18(1) (2011) P109, 24 pages.

Cf. A192090, A192092, A033508 (without tatami condition). Row sums of A272474.

A281791 Ways to tile a 5 X (2n+1) floor with tatami mats, including one monomer.

Original entry on oeis.org

3, 18, 10, 8, 18, 24, 32, 52, 68, 100, 142, 196, 280, 388, 542, 756, 1046, 1452, 2006, 2768, 3816, 5248, 7212, 9896, 13562, 18568, 25392, 34692, 47354, 64580, 88002, 119824, 163034, 221672, 301200, 409004, 555060, 752844, 1020550, 1382732, 1872520, 2534596, 3429206, 4637556, 6269070

Offset: 0

Yasutoshi Kohmoto, Jan 30 2017

Apart from a single 1 X 1 monomer, the area is tiled with 2 X 1 mats. No four mats are permitted to meet at a point.

			For n=0, the 5X1 floor allows the monomer to be placed at one of the two ends or in the middle: a(n=0)=3.

Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-2,-1).

Cf. A271786 [3X(2n+1) floor]. 2nd column of A272474.

s1(n)=my(s); forstep(k=(n%4!=1),(n-1)\6,2, s+=((n+3)/4-k/2)*((n-1)/4-k/2)!/(k!*((n-1)/4-3*k/2)!)); 2*s
s3(n)=my(s); forstep(k=(n%4==1),(n-3)\6,2, s+=((n-3)/4-k/2)!/(k!*((n-3)/4-3*k/2)!)); 2*s
s5(n)=my(s); forstep(k=(n%4!=1),(n-5)\6,2, s+=((n+7)/4-k/2)*((n-5)/4-k/2)!/(k!*((n-5)/4-3*k/2)!)); 2*s
a(n)=s1(n) + s3(n) + s5(n) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = S_1(2n+1) + S_5(2n+1) + S_3(2n+1) for n>1 where
S_1(n) = 2* Sum_{k= 0<=k<=[(n-1)/6]} ((n+3)/4-1/2*k) *((n-1)/4-1/2*k)!/(k!*((n-1)/4-3/2*k)!). The sum is over even k if n==1 (mod 4), else over odd k.
S_5(n) = 2* Sum_{0<=k<=[(n-5)/6]} ((n+7)/4-1/2*k) *((n-5)/4-1/2*k)!/(k!*((n-5)/4-3/2*k)!). The sum is over even k if n==1 (mod 4) else over odd k.
S_3(n) = 2* Sum_{0<=k<=[(n-3)/6]} 2*((n-3)/4-1/2*k)!/(k!*((n-3)/4-3/2*k)!). The sum is over odd k if n==1 (mod 4), else over even k.
Where [m] is floor(m).
G.f. x +14*x^3 +2*x*(1 +2*x^2 +3*x^4 -2*x^6 -4*x^8 -2*x^10)/ (1-x^4-x^6)^2. (Includes zeros for even floor widths).- R. J. Mathar, Apr 10 2017
a(n) = 2*(A228577(n-1)+A228577(n+1))+4*(A182097(n-2)+A182097(n-1)), n>1. - R. J. Mathar, Apr 10 2017

cp's OEIS Frontend

A068924 Number of ways to tile a 5 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Views

Author

Keywords

Links

Crossrefs

Formula

A192091 Number of tatami tilings of a 5 X n grid (with monomers allowed).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A281791 Ways to tile a 5 X (2n+1) floor with tatami mats, including one monomer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula