A067925 Consider a room of size r X s where rs = 2n and 1 <= r, 1 <= 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 distinguished if one is a rotation or reflection of the other.

2, 4, 8, 10, 14, 28, 28, 42, 70, 90, 122, 204, 260, 386, 592, 824, 1192, 1810, 2558, 3764, 5580, 8064, 11794, 17438, 25338, 37144, 54626, 79762, 116852, 171650, 250984, 367874, 539668, 790110, 1157912, 1697978, 2487050, 3645012, 5343444

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..39.

Dean Hickerson, Filling rectangular rooms with Tatami mats

Example

For n=3 there are 3 incongruent tilings, shown below. These can be rotated to produce 8 tilings, so a(3)=8.
._____. ._____.
|___| | | | | | .___________.
|___|_| |_|_|_| |___|___|___|

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]];
A067925[n_] := Module[{i, divs}, divs = Divisors[2 n]; Sum[t[divs[[i]], 2 n/divs[[i]]], {i, 1, Length[divs]}]];
Table[A067925[n], {n, 1, 50}] (* Jean-François Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)

Crossrefs

Cf. A052270 for number of incongruent tilings, A068920 for table of number of tilings of an r X s room.

Sequence in context: A089033 A049422 A178215 * A323102 A331627 A102431

Adjacent sequences: A067922 A067923 A067924 * A067926 A067927 A067928

Keyword

nonn,nice

Author

Yasutoshi Kohmoto Mar 05 2002

Extensions

Edited by Dean Hickerson, Mar 11 2002

Status

approved