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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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..42.

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

EXAMPLE

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:

._____. ._____.

|___| | | | | | .___________.

|___|_| |_|_|_| |___|___|___|

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-Fran├žois Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)

CROSSREFS

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

Sequence in context: A284311 A174225 A182945 * A265335 A179223 A069117

Adjacent sequences:  A052267 A052268 A052269 * A052271 A052272 A052273

KEYWORD

nonn,nice

AUTHOR

Yasutoshi Kohmoto

EXTENSIONS

Extended by Dean Hickerson, Mar 01 2002

STATUS

approved

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Last modified April 20 16:17 EDT 2019. Contains 322310 sequences. (Running on oeis4.)