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

Author

Yasutoshi Kohmoto Mar 05 2002

Keywords

• nonn
• nice

Comments

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

Examples

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

Links

• Dean Hickerson, Filling rectangular rooms with Tatami mats

Crossrefs

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

Programs

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 *)

Extensions

Edited by Dean Hickerson, Mar 11 2002