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A068920 Table of t(r,s) read by antidiagonals: t(r,s) is the number of ways to tile an r X s room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

%I A068920 #26 Mar 08 2022 03:05:35
%S A068920 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 A068920 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 A068920 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 A068920 Table of t(r,s) read by antidiagonals: t(r,s) is the number of ways to tile an r X s room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.
%C A068920 Rows 2-6 are given in A068921 - A068925.
%H A068920 Dean Hickerson, <a href="/A068920/a068920.txt">Filling rectangular rooms with Tatami mats</a>
%e A068920 Table begins:
%e A068920   0, 1, 0, 1, 0, 1, ...
%e A068920   1, 2, 3, 4, 6, 9, ...
%e A068920   0, 3, 0, 4, 0, 6, ...
%e A068920   1, 4, 4, 2, 3, 3, ...
%e A068920   0, 6, 0, 3, 0, 2, ...
%e A068920   1, 9, 6, 3, 2, 2, ...
%e A068920   ...
%t A068920 (* See link for Mathematica programs. *)
%t A068920 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 A068920 c1[r_, s_] := Which[s <= 0, 0, r == 2, c[2, s - 1], EvenQ[r], c2[r, s - 1] + Boole[s == 1]];
%t A068920 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 A068920 t[r_, s_] := Which[r>s, t[s, r], OddQ[r] && r>1, 2 c[r, s], True, c[r, s]];
%t A068920 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 A068920 Table[A068920[n], {n, 0, 100}] (* _Jean-François Alcover_, May 12 2017, copied and adapted from _Dean Hickerson_'s programs *)
%Y A068920 Cf. A068926 for incongruent tilings, A067925 for count by area.
%Y A068920 Cf. A068921 (row 2), A068922 (row 3), A068923 (row 4), A068924 (row 5), A068925 (row 6).
%K A068920 nonn,tabl
%O A068920 1,5
%A A068920 _Dean Hickerson_, Mar 11 2002