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.

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

Author

Yasutoshi Kohmoto

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 2 ways to cover a 2 X 3 room and 1 way to cover a 1 X 6 room, so a(3)=3:
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Links

• 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

Crossrefs

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

Programs

Mathematica

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 *)

Extensions

Extended by Dean Hickerson, Mar 01 2002