%I
%S 0,1,1,0,2,0,1,3,3,1,0,4,0,4,0,1,6,4,4,6,1,0,9,0,2,0,9,0,1,13,6,3,3,6,
%T 13,1,0,19,0,3,0,3,0,19,0,1,28,10,3,2,2,3,10,28,1,0,41,0,5,0,2,0,5,0,
%U 41,0,1,60,16,5,2,2,2,2,5,16,60,1,0,88,0,6,0,1,0,1,0,6,0,88,0,1,129,26
%N Table of t(r,s) by diagonals, where t(r,s) is the number of ways to tile an r X s room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.
%C Table begins: t(1,1); t(1,2) t(2,1); t(1,3) t(2,2) t(3,1); ... Rows 26 are given in A068921  A068925.
%H Dean Hickerson, <a href="/A068920/a068920.txt">Filling rectangular rooms with Tatami mats</a>
%t (* See link for Mathematica programs. *)
%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 t[r_, s_] := Which[r>s, t[s, r], OddQ[r] && r>1, 2 c[r, s], True, c[r, s]];
%t A068920[n_] := Module[{x}, x = Floor[(Sqrt[8 n + 1]  1)/2]; t[n + 1  x (x + 1)/2, (x + 1) (x + 2)/2  n]];
%t Table[A068920[n], {n, 0, 100}] (* _JeanFrançois Alcover_, May 12 2017, copied and adapted from _Dean Hickerson_'s programs *)
%Y Cf. A068926 for incongruent tilings, A067925 for count by area, A068921 for row 2, A068922 for row 3, A068923 for row 4, A068924 for row 5, A068925 for row 6.
%K nonn,tabl
%O 1,5
%A _Dean Hickerson_, Mar 11 2002
