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A068926 Table of ti(r,s) by diagonals, where ti(r,s) is the number of incongruent ways to tile an r X s room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point. 8
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, 1, 69, 9, 5, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Table begins: ti(1,1); ti(1,2) ti(2,1); ti(1,3) ti(2,2) ti(3,1); ... Rows 2-6 are given in A068927 - A068931.

LINKS

Table of n, a(n) for n=0..96.

Dean Hickerson, Filling rectangular rooms with Tatami mats

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-Fran├žois Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)

CROSSREFS

Cf. A068920 for total number of tilings, A052270 for count by area, A068927 for row 2, A068928 for row 3, A068929 for row 4, A068930 for row 5, A068931 for row 6.

Sequence in context: A177975 A062135 A190182 * A276770 A258291 A146527

Adjacent sequences:  A068923 A068924 A068925 * A068927 A068928 A068929

KEYWORD

nonn,tabl

AUTHOR

Dean Hickerson, Mar 11 2002

STATUS

approved

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Last modified April 25 04:08 EDT 2019. Contains 322450 sequences. (Running on oeis4.)