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A068920
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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.
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13
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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, 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, 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
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OFFSET
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1,5
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COMMENTS
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Table begins: t(1,1); t(1,2) t(2,1); t(1,3) t(2,2) t(3,1); ... Rows 2-6 are given in A068921 - A068925.
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LINKS
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Table of n, a(n) for n=1..94.
Dean Hickerson, Filling rectangular rooms with Tatami mats
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MATHEMATICA
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(* See link 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]];
t[r_, s_] := Which[r>s, t[s, r], OddQ[r] && r>1, 2 c[r, s], True, c[r, s]];
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]];
Table[A068920[n], {n, 0, 100}] (* Jean-François Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)
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CROSSREFS
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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.
Sequence in context: A291556 A323073 A167279 * A099390 A297477 A124031
Adjacent sequences: A068917 A068918 A068919 * A068921 A068922 A068923
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KEYWORD
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nonn,tabl
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AUTHOR
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Dean Hickerson, Mar 11 2002
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STATUS
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approved
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