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%I #14 May 12 2017 12:02:37
%S 1,2,3,4,5,9,9,14,19,27,34,56,70,105,152,218,308,466,654,966,1407,
%T 2052,2979,4399,6378,9361,13697,20051,29308,43035,62885,92204,135053,
%U 197871,289775,424891,622199,911988,1336319,1958344,2869418,4205888
%N 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.
%C Tatami mats are of size 1 X 2; at most 3 may meet at a point.
%H Dean Hickerson, <a href="/A052270/a052270.txt">Illustration of first few cases</a>
%H Dean Hickerson, <a href="/A068920/a068920.txt">Filling rectangular rooms with Tatami mats</a> (Includes Mathematica program)
%H Yasutoshi Kohmoto, <a href="/A052270/a052270.gif">Illustration of a(6) = 9</a>
%e 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:
%e ._____. ._____.
%e |___| | | | | | .___________.
%e |___|_| |_|_|_| |___|___|___|
%t 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 c1[r_, s_] := Which[s <= 0, 0, r == 2, c[2, s - 1], EvenQ[r], c2[r, s - 1] + Boole[s == 1]];
%t 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 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]];
%t 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]];
%t 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]];
%t ti[r_, s_] := Which[r > s, ti[s, r], r == s, 1 - Mod[r, 2], True, (c[r, s] + cs[r, s])/2];
%t A052270[n_] := Module[{i, divs}, divs = Divisors[2 n]; Sum[ti[divs[[i]], 2 n/divs[[i]]], {i, 1, Ceiling[Length[divs]/2]}]];
%t Table[A052270[n], {n, 1, 50}] (* _Jean-François Alcover_, May 12 2017, copied and adapted from _Dean Hickerson_'s programs *)
%Y Cf. A067925 for total number of tilings, A068926 for table of number of incongruent tilings of an r X s room.
%K nonn,nice
%O 1,2
%A _Yasutoshi Kohmoto_
%E Extended by _Dean Hickerson_, Mar 01 2002
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