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%I #25 Mar 08 2022 03:03:28
%S 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,
%T 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,
%U 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
%N 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.
%H Michael De Vlieger, <a href="/A068926/b068926.txt">Table of n, a(n) for n = 0..11325</a> (antidiagonals 1..150, flattened)
%H Dean Hickerson, <a href="/A068920/a068920.txt">Filling rectangular rooms with Tatami mats</a>
%e Table begins:
%e 0, 1, 0, 1, 0, 1, 0, ...
%e 1, 1, 2, 3, 4, 6, 8, ...
%e 0, 2, 0, 2, 0, 2, 0, ...
%e 1, 3, 2, 1, 2, 2, 2, ...
%e 0, 4, 0, 2, 0, 1, 0, ...
%e 1, 6, 2, 2, 1, 1, 1, ...
%e 0, 8, 0, 2, 0, 1, 0, ...
%e ...
%t (* See link above 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 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 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]];
%t Table[A068926[n], {n, 0, 100}] (* _Jean-François Alcover_, May 12 2017, copied and adapted from _Dean Hickerson_'s programs *)
%Y Cf. A068920 (total number of tilings), A052270 (count by area).
%Y Cf. A068927 (row 2), A068928 (row 3), A068929 (row 4), A068930 (row 5), A068931 (row 6).
%K nonn,tabl
%O 0,8
%A _Dean Hickerson_, Mar 11 2002
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