A068923 -id:A068923

Search: a068923 -id:a068923

Displaying 1-3 of 3 results found.

page 1

A068920

Table of t(r,s) read by antidiagonals: t(r,s) is the number of ways to tile an r X s room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

+10
13

0, 1, 1, 0, 2, 0, 1, 3, 3, 1, 0, 4, 0, 4, 0, 1, 6, 4, 4, 6, 1, 0, 9, 0, 2, 0, 9, 0, 1, 13, 6, 3, 3, 6, 13, 1, 0, 19, 0, 3, 0, 3, 0, 19, 0, 1, 28, 10, 3, 2, 2, 3, 10, 28, 1, 0, 41, 0, 5, 0, 2, 0, 5, 0, 41, 0, 1, 60, 16, 5, 2, 2, 2, 2, 5, 16, 60, 1, 0, 88, 0, 6, 0, 1, 0, 1, 0, 6, 0, 88, 0, 1, 129, 26

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Rows 2-6 are given in A068921 - A068925.

LINKS

Table of n, a(n) for n=1..94.

Dean Hickerson, Filling rectangular rooms with Tatami mats

EXAMPLE

Table begins:

0, 1, 0, 1, 0, 1, ...

1, 2, 3, 4, 6, 9, ...

0, 3, 0, 4, 0, 6, ...

1, 4, 4, 2, 3, 3, ...

0, 6, 0, 3, 0, 2, ...

1, 9, 6, 3, 2, 2, ...

...

MATHEMATICA

(* See link for Mathematica programs. *)

c[r_, s_] := Which[s<0, 0, r==1, 1 - Mod[s, 2], r == 2, c1[2, s] + c2[2, s] + Boole[s == 0], OddQ[r], c[r, s] = c[r, s - r + 1] + c[r, s - r - 1] + Boole[s == 0], EvenQ[r], c[r, s] = c1[r, s] + c2[r, s] + Boole[s == 0]];

c1[r_, s_] := Which[s <= 0, 0, r == 2, c[2, s - 1], EvenQ[r], c2[r, s - 1] + Boole[s == 1]];

c2[r_, s_] := Which[s <= 0, 0, r == 2, c2[2, s] = c1[2, s - 2] + Boole[s == 2], EvenQ[r], c2[r, s] = c1[r, s - r + 2] + c1[r, s - r] + Boole[s == r - 2] + Boole[s == r]];

t[r_, s_] := Which[r>s, t[s, r], OddQ[r] && r>1, 2 c[r, s], True, c[r, s]];

A068920[n_] := Module[{x}, x = Floor[(Sqrt[8 n + 1] - 1)/2]; t[n + 1 - x (x + 1)/2, (x + 1) (x + 2)/2 - n]];

Table[A068920[n], {n, 0, 100}] (* Jean-François Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)

CROSSREFS

Cf. A068926 for incongruent tilings, A067925 for count by area.

Cf. A068921 (row 2), A068922 (row 3), A068923 (row 4), A068924 (row 5), A068925 (row 6).

KEYWORD

nonn,tabl

AUTHOR

Dean Hickerson, Mar 11 2002

STATUS

approved

A272473

Triangle T(n,m) by rows: the number of tatami tilings of a 4 by n grid with 2*m monomers.

+10
3

1, 3, 1, 4, 18, 7, 4, 27, 13, 2, 32, 32, 3, 52, 64, 7, 3, 62, 133, 40, 3, 99, 269, 110, 9, 5, 152, 437, 280, 48, 5, 163, 730, 669, 138, 9, 6, 258, 1243, 1318, 433, 48, 8, 343, 1823, 2670, 1239, 154, 9, 8, 408, 2949, 5240, 2849, 600, 48, 11, 632, 4577

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The number of squares in the 4 by n floor is even, so the number of tilings with an odd number of monomers is zero.

LINKS

Table of n, a(n) for n=1..59.

FORMULA

G.f. x*( -1 -8*x^7*y^2 +21*x^5*y^2 -7*x^7*y^6 +4*x^3*y^2 -3*x^7 +2*x^5 -8*x^2*y^2 -4*x^8*y^4 -3*x -6*x*y^4 -15*x*y^2 -2*x^3*y^4 -6*x^8 -5*x^10*y^2 -5*x^9*y^2 -y^4 -2*x^8*y^2 -3*y^2 -8*x^11*y^2 +5*x^11*y^4 -3*x^2*y^4 -2*x^5*y^6 +2*x^13 +x^12 +x^11 +x^6 -7*x^7*y^4 +x^7*y^8 +11*x^4*y^2 -3*x^9 -15*x^10*y^4 -2*x^10*y^6 +18*x^9*y^4 +36*x^6*y^4 +20*x^6*y^2 -17*x^ 5*y^4 -8*x^4*y^4 +4*x^3 +8*x^6*y^6 +5*x^4 +2*x^9*y^6 -y^8*x^6 +6*y^6*x^3 +y^6*x^2)/ (x^11 -x^10 +2*x^9 -3*x^9*y^2 +x^8*y^2 -2*x^8 +x^7 +x^6*y^4 -5*x^6*y^2 -3*x^6 +2*x^5 +5*x^5*y^2 +x^4*y^2 -2*x^4 -x^3*y^2 +2*x^3 +x^2*y^2 +x -1). - R. J. Mathar, May 01 2016

EXAMPLE

The triangle starts in row n=1 and column m=0 as:

1,3,1;

4,18,7;

4,27,13;

2,32,32;

3,52,64,7;

3,62,133,40;

3,99,269,110,9;

5,152,437,280,48;

5,163,730,669,138,9;

6,258,1243,1318,433,48;

8,343,1823,2670,1239,154,9;

8,408,2949,5240,2849,600,48;

11,632,4577,9011,6655,1927,172,9;

13,746,6287,16184,14697,4930,777,48;

14,971,9928,28135,28805,13089,2669,190,9;

19,1394,14234,44806,58022,32176,7501,954,48;

21,1610,19501,75702,111795,70427,22344,3445,208,9;

25,2224,29785,121302,199354,157078,59859,10576,1131,48;

32,2909,40073,184597,366553,331449,143611,34646,4257,226,9;

35,3464,55939,298278,644436,651772,350855,99300,14167,1308,48;

44,4820,81474,449995,1081033,1303651,802565,258303,50095,5105,244,9;

53,5924,106460,670726,1868914,2488996,1719501,684338,151835,18274,1485,48;

60,7408,150672,1040424,3077401,4548409,3716945,1678785,425017,68761,5989,262,9;

76,9972,208211,1503372,4956628,8434302,7641320,3879356,1208052,218806,22897,1662,48;

CROSSREFS

Cf. A192090 (row sums), A068923 (column m=0), A272472 (3 by n grid), A100265 (without tatami condition, reversed rows).

KEYWORD

nonn,tabf

AUTHOR

R. J. Mathar, Apr 30 2016

STATUS

approved

A068929

Number of incongruent ways to tile a 4 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

+10
2

1, 3, 2, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 11, 12, 14, 17, 20, 24, 29, 32, 41, 46, 56, 68, 78, 93, 114, 130, 161, 188, 223, 268, 318, 378, 456, 533, 646, 763, 911, 1092, 1296, 1542, 1855, 2190, 2634, 3133, 3732, 4463, 5323, 6339, 7596, 9022, 10802, 12876

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..57.

F. Ruskey and J. Woodcock, Counting Fixed-Height Tatami Tilings, Electronic Journal of Combinatorics, Paper R126 (2009) 20 pages.

FORMULA

For n >= 20, a(n) = a(n-3) + a(n-5) + a(n-6) - a(n-9) + a(n-10) - a(n-11) - a(n-13) - a(n-15).

G.f.: x*(1-x^18+x^17+x^16+x^15+x^13-x^12-2*x^11-2*x^8-4*x^7-3*x^6-x^5-x^4+2*x^2+3*x) / ((x^5+x^3-1) * (x^10+x^6-1)) [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]

CROSSREFS

Cf. A068923 for total number of tilings, A068926 for more info.

KEYWORD

easy,nonn

AUTHOR

Dean Hickerson, Mar 11 2002

EXTENSIONS

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.

STATUS

approved

page 1

Search completed in 0.005 seconds