A068926 Table read by antidiagonals: ti(r,s) is the number of incongruent ways to tile an r X s room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

0, 1, 1, 0, 1, 0, 1, 2, 2, 1, 0, 3, 0, 3, 0, 1, 4, 2, 2, 4, 1, 0, 6, 0, 1, 0, 6, 0, 1, 8, 2, 2, 2, 2, 8, 1, 0, 12, 0, 2, 0, 2, 0, 12, 0, 1, 16, 4, 2, 1, 1, 2, 4, 16, 1, 0, 24, 0, 3, 0, 1, 0, 3, 0, 24, 0, 1, 33, 5, 3, 1, 1, 1, 1, 3, 5, 33, 1, 0, 49, 0, 4, 0, 1, 0, 1, 0, 4, 0, 49, 0

Offset: 0

Author

Dean Hickerson, Mar 11 2002

Keywords

• nonn
• tabl

Examples

			Table begins:
  0, 1, 0, 1, 0, 1, 0, ...
  1, 1, 2, 3, 4, 6, 8, ...
  0, 2, 0, 2, 0, 2, 0, ...
  1, 3, 2, 1, 2, 2, 2, ...
  0, 4, 0, 2, 0, 1, 0, ...
  1, 6, 2, 2, 1, 1, 1, ...
  0, 8, 0, 2, 0, 1, 0, ...
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11325 (antidiagonals 1..150, flattened)
• Dean Hickerson, Filling rectangular rooms with Tatami mats

Crossrefs

Cf. A068920 (total number of tilings), A052270 (count by area).

Cf. A068927 (row 2), A068928 (row 3), A068929 (row 4), A068930 (row 5), A068931 (row 6).

Programs

Mathematica

(* See link above 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]];
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]];
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]];
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]];
ti[r_, s_] := Which[r > s, ti[s, r], r == s, 1 - Mod[r, 2], True, (c[r, s] + cs[r, s])/2];
A068926[n_] := Module[{x}, x = Floor[(Sqrt[8 n + 1] - 1)/2]; ti[n + 1 - x (x + 1)/2, (x + 1) (x + 2)/2 - n]];
Table[A068926[n], {n, 0, 100}] (* Jean-François Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)