The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A361221 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle, up to rotations and reflections. 7
 1, 1, 1, 1, 5, 8, 2, 12, 95, 719, 2, 31, 682, 20600, 315107 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 LINKS Table of n, a(n) for n=1..15. EXAMPLE Triangle begins: n\k| 1 2 3 4 5 ---+------------------------ 1 | 1 2 | 1 1 3 | 1 5 8 4 | 2 12 95 719 5 | 2 31 682 20600 315107 A 3 X 3 square can be tiled by one 1 X 3 piece, two 1 X 2 pieces and two 1 X 1 pieces in the following 8 ways: +---+---+---+ +---+---+---+ +---+---+---+ | | | | | | | | | | +---+---+ + +---+---+---+ +---+---+---+ | | | | | | | | | +---+---+---+ +---+---+---+ +---+---+---+ | | | | | | +---+---+---+ +---+---+---+ +---+---+---+ . +---+---+---+ +---+---+---+ +---+---+---+ | | | | | | | | | | | + + +---+ + +---+ + + +---+---+ | | | | | | | | | | | | +---+---+---+ +---+---+---+ +---+---+---+ | | | | | | +---+---+---+ +---+---+---+ +---+---+---+ . +---+---+---+ +---+---+---+ | | | | | | +---+---+---+ +---+---+---+ | | | | +---+---+---+ +---+---+---+ | | | | | | +---+---+---+ +---+---+---+ This is the maximum for a 3 X 3 square, so T(3,3) = 8. There is one other set of pieces that also can tile the 3 X 3 square in 8 ways: three 1 X 2 pieces and three 1 X 1 pieces (see illustration in A361216). The following table shows all sets of pieces that give the maximum number of tilings for 1 <= k <= n <= 5: \ Number of pieces of size (n,k)\ 1 X 1 | 1 X 2 | 1 X 3 | 2 X 2 ------+-------+-------+-------+------ (1,1) | 1 | 0 | 0 | 0 (2,1) | 2 | 0 | 0 | 0 (2,1) | 0 | 1 | 0 | 0 (2,2) | 4 | 0 | 0 | 0 (2,2) | 2 | 1 | 0 | 0 (2,2) | 0 | 2 | 0 | 0 (2,2) | 0 | 0 | 0 | 1 (3,1) | 3 | 0 | 0 | 0 (3,1) | 1 | 1 | 0 | 0 (3,1) | 0 | 0 | 1 | 0 (3,2) | 2 | 2 | 0 | 0 (3,3) | 3 | 3 | 0 | 0 (3,3) | 2 | 2 | 1 | 0 (4,1) | 2 | 1 | 0 | 0 (4,2) | 4 | 2 | 0 | 0 (4,3) | 3 | 3 | 1 | 0 (4,4) | 5 | 4 | 1 | 0 (5,1) | 3 | 1 | 0 | 0 (5,1) | 2 | 0 | 1 | 0 (5,1) | 1 | 2 | 0 | 0 (5,2) | 4 | 3 | 0 | 0 (5,3) | 4 | 4 | 1 | 0 (5,4) | 7 | 5 | 1 | 0 (5,5) | 7 | 6 | 2 | 0 It seems that all optimal solutions for A361216 are also optimal here, but occasionally there are other optimal solutions, e.g. for n = k = 3. CROSSREFS Main diagonal: A361222. Columns: A361223 (k = 1), A361224 (k = 2), A361225 (k = 3). Cf. A360629, A361216. Sequence in context: A198844 A156035 A284697 * A319261 A010489 A196567 Adjacent sequences: A361218 A361219 A361220 * A361222 A361223 A361224 KEYWORD nonn,tabl,more AUTHOR Pontus von Brömssen, Mar 05 2023 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 18 13:50 EDT 2024. Contains 371780 sequences. (Running on oeis4.)