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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A102431 A076919

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

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Last modified April 24 22:03 EDT 2019. Contains 322446 sequences. (Running on oeis4.)