A068926 -id:A068926 - cp's OEIS frontend

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.

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

Offset: 1

Dean Hickerson, Mar 11 2002

Rows 2-6 are given in A068921 - A068925.

			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, ...
  ...

Dean Hickerson, Filling rectangular rooms with Tatami mats

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).

(* 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 *)

A068928 Number of incongruent ways to tile a 3 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

2, 2, 2, 4, 5, 9, 12, 21, 30, 51, 76, 127, 195, 322, 504, 826, 1309, 2135, 3410, 5545, 8900, 14445, 23256, 37701, 60813, 98514, 159094, 257608, 416325, 673933, 1089648, 1763581, 2852242, 4615823, 7466468, 12082291, 19546175, 31628466

Offset: 1

Dean Hickerson, Mar 11 2002

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 10.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,0,-1,-1).

Cf. A068922 for total number of tilings, A068926 for more info.

Essentially the same as A001224.

For n >= 8, a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-5) - a(n-6).

O.g.f.: x(2-4x^2-x^4+x^6)/((1-x-x^2)(1-x^2-x^4)). a(n) = (A000045(n+1)+A053602(n+1))/2, n>1. [From R. J. Mathar, Aug 30 2008]

A068930 Number of incongruent ways to tile a 5 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

4, 2, 1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 9, 13, 15, 22, 26, 37, 45, 63, 78, 108, 136, 186, 237, 322, 414, 559, 724, 973, 1267, 1697, 2219, 2964, 3888, 5183, 6815, 9071, 11949, 15886, 20955, 27835, 36755, 48790, 64476, 85545, 113115, 150021, 198460, 263136

Offset: 1

Dean Hickerson, Mar 11 2002

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 12.
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -1).

Cf. A068924 for total number of tilings, A068926 for more info.

Cf. A005683.

Join[{4,2},LinearRecurrence[{0,1,1,1,0,0,-1,-1,-1},{1,1,1,2,2,3,3,5,5},50]] (* Harvey P. Dale, Nov 21 2014 *)

For n >= 12, a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-7) - a(n-8) - a(n-9).
G.f.: x*(4+x^10+5*x^9+4*x^8+3*x^7-x^6-2*x^5-6*x^4-5*x^3 -3*x^2+2*x) / ((x^3+x^2-1)*(x^6+x^4-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
a(n) = sum(A102541(n-k-2, n-2*k-4), k=0..floor((n-4)/2)), n >= 4. - Johannes W. Meijer, Aug 24 2013

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

A052270 Consider a room of size r X s where rs = 2n and 1 <= r <= s; count ways to arrange n Tatami mats in room; a(n) = total number of ways for all choices of r and s. Two arrangements are considered the same if one is a rotation or reflection of the other.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 9, 14, 19, 27, 34, 56, 70, 105, 152, 218, 308, 466, 654, 966, 1407, 2052, 2979, 4399, 6378, 9361, 13697, 20051, 29308, 43035, 62885, 92204, 135053, 197871, 289775, 424891, 622199, 911988, 1336319, 1958344, 2869418, 4205888

Offset: 1

Yasutoshi Kohmoto

Tatami mats are of size 1 X 2; at most 3 may meet at a point.

			For n = 3 there are 2 ways to cover a 2 X 3 room and 1 way to cover a 1 X 6 room, so a(3)=3:
._____. ._____.
|___| | | | | | .___________.
|___|_| |_|_|_| |___|___|___|

Dean Hickerson, Illustration of first few cases
Dean Hickerson, Filling rectangular rooms with Tatami mats (Includes Mathematica program)
Yasutoshi Kohmoto, Illustration of a(6) = 9

Cf. A067925 for total number of tilings, A068926 for table of number of incongruent tilings of an r X s room.

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]];
cs[r_, s_] := Which[s < 0, 0, r == 1, c[r, s], r == 2, cs[2, s] = c1s[r, s] + c2s[r, s] + Boole[s == 0], OddQ[r], cs[r, s] = cs[r, s - 2 r + 2] + cs[r, s - 2 r - 2] + Boole[s == 0] + Boole[s == r - 1] + Boole[s == r + 1], EvenQ[r], cs[r, s] = c1s[r, s] + c2s[r, s] + Boole[s == 0]];
c1s[r_, s_] := Which[s <= 0, 0, r == 2, cs[r, s - 2] + Boole[s == 1], EvenQ[r], c2s[r, s - 2] + Boole[s == 1]];
c2s[r_, s_] := Which[s <= 0, 0, r == 2, c2s[2, s] = c1s[2, s - 4] + Boole[s == 2], EvenQ[r], c2s[r, s] = c1s[r, s - 2 r + 4] + c1s[r, s - 2 r] + Boole[s == r - 2] + Boole[s == r]];
ti[r_, s_] := Which[r > s, ti[s, r], r == s, 1 - Mod[r, 2], True, (c[r, s] + cs[r, s])/2];
A052270[n_] := Module[{i, divs}, divs = Divisors[2 n]; Sum[ti[divs[[i]], 2 n/divs[[i]]], {i, 1, Ceiling[Length[divs]/2]}]];
Table[A052270[n], {n, 1, 50}] (* Jean-François Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)

Extended by Dean Hickerson, Mar 01 2002

A068927 Number of incongruent ways to tile a 2 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 12, 16, 24, 33, 49, 69, 102, 145, 214, 307, 452, 653, 960, 1393, 2046, 2978, 4371, 6376, 9354, 13665, 20041, 29307, 42972, 62884, 92191, 134974, 197858, 289772, 424746, 622198, 911970, 1336121, 1958319, 2869417, 4205538

Offset: 1

Dean Hickerson, Mar 11 2002

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 9.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,1,-1,0,-1).

Cf. A068921 for total number of tilings, A068926 for more info.

For n >= 12, a(n) = a(n-1) + a(n-2) - a(n-5) + a(n-6) - a(n-7) - a(n-9).

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

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

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.

Original entry on oeis.org

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

Offset: 1

Dean Hickerson, Mar 11 2002

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

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

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]

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

A068931 Number of incongruent ways to tile a 6 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

1, 6, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 6, 7, 7, 8, 8, 10, 12, 13, 14, 15, 17, 20, 21, 26, 26, 31, 34, 38, 44, 47, 56, 60, 66, 78, 82, 100, 104, 122, 134, 148, 176, 186, 217, 238, 266, 310, 328, 393, 417, 483, 543, 594, 694, 745, 870, 960, 1066, 1237

Offset: 1

Dean Hickerson, Mar 11 2002

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 13.
F. Ruskey and J. Woodcock, Counting Fixed-Height Tatami Tilings, Electronic Journal of Combinatorics, Paper R126 (2009) 20 pages. [_Frank Ruskey_, Sep 26 2010]

Cf. A068925 for total number of tilings, A068926 for more info.

For n >= 28, a(n) = a(n-5) + a(n-7) + a(n-10) + a(n-14) - a(n-15) - a(n-17) - a(n-19) - a(n-21).

G.f.: -x*(-1-6*x-2*x^2-3*x^16+6*x^15 +2*x^14-6*x^18 +2*x^7+7*x^8+2*x^9 +2*x^10 +6*x^11 +2*x^12+2*x^13-2*x^19-5*x^20-2*x^21 -6*x^22-2*x^23 +x^26 -2*x^3-x^4 +5*x^6)/ ((x^2-x+1) * (x^5+x^4+x^3-x-1) * (x^4-x^2+1) * (x^10 +x^8 +x^6-x^2-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by R. J. Mathar, Sep 16 2009

cp's OEIS Frontend

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.

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A068928 Number of incongruent ways to tile a 3 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

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A068930 Number of incongruent ways to tile a 5 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

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A052270 Consider a room of size r X s where rs = 2n and 1 <= r <= s; count ways to arrange n Tatami mats in room; a(n) = total number of ways for all choices of r and s. Two arrangements are considered the same if one is a rotation or reflection of the other.

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A068927 Number of incongruent ways to tile a 2 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

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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.

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A068931 Number of incongruent ways to tile a 6 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

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