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

0,8

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

Example

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, ...
  ...

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 *)

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).

Sequence in context: A062135 A190182 A371788 * A276770 A258291 A146527

Adjacent sequences: A068923 A068924 A068925 * A068927 A068928 A068929

Keyword

nonn,tabl

Author

Dean Hickerson, Mar 11 2002

Status

approved