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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.

%I A067925 #13 May 18 2019 16:59:10
%S A067925 2,4,8,10,14,28,28,42,70,90,122,204,260,386,592,824,1192,1810,2558,
%T A067925 3764,5580,8064,11794,17438,25338,37144,54626,79762,116852,171650,
%U A067925 250984,367874,539668,790110,1157912,1697978,2487050,3645012,5343444
%N 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.
%C A067925 Tatami mats are of size 1 X 2; at most 3 may meet at a point.
%H A067925 Dean Hickerson, <a href="/A068920/a068920.txt">Filling rectangular rooms with Tatami mats</a>
%e A067925 For n=3 there are 3 incongruent tilings, shown below. These can be rotated to produce 8 tilings, so a(3)=8.
%e A067925 ._____. ._____.
%e A067925 |___| | | | | | .___________.
%e A067925 |___|_| |_|_|_| |___|___|___|
%t A067925 (* See link for Mathematica programs. *)
%t A067925 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]];
%t A067925 c1[r_, s_] := Which[s <= 0, 0, r == 2, c[2, s - 1], EvenQ[r], c2[r, s - 1] + Boole[s == 1]];
%t A067925 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 A067925 t[r_, s_] := Which[r>s, t[s, r], OddQ[r] && r>1, 2 c[r, s], True, c[r, s]];
%t A067925 A067925[n_] := Module[{i, divs}, divs = Divisors[2 n]; Sum[t[divs[[i]], 2 n/divs[[i]]], {i, 1, Length[divs]}]];
%t A067925 Table[A067925[n], {n, 1, 50}] (* _Jean-François Alcover_, May 12 2017, copied and adapted from _Dean Hickerson_'s programs *)
%Y A067925 Cf. A052270 for number of incongruent tilings, A068920 for table of number of tilings of an r X s room.
%K A067925 nonn,nice
%O A067925 1,1
%A A067925 _Yasutoshi Kohmoto_ Mar 05 2002
%E A067925 Edited by _Dean Hickerson_, Mar 11 2002