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 A362142 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, 0 <= k <= n. 6
 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 3, 6, 16, 1, 1, 4, 12, 37, 140, 1, 1, 6, 24, 105, 454, 1987, 1, 1, 10, 40, 250, 1566, 9856, 62266, 1, 1, 15, 80, 726, 5670, 47394, 406168, 3899340, 1, 1, 21, 160, 1824, 18738, 223696, 2916492, 38322758, 508317004 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Pontus von Brömssen, Table of n, a(n) for n = 0..90 (rows 0..12) EXAMPLE Triangle begins: n\k| 0 1 2 3 4 5 6 7 8 ---+----------------------------------------- 0 | 1 1 | 1 1 2 | 1 1 1 3 | 1 1 2 4 4 | 1 1 3 6 16 5 | 1 1 4 12 37 140 6 | 1 1 6 24 105 454 1987 7 | 1 1 10 40 250 1566 9856 62266 8 | 1 1 15 80 726 5670 47394 406168 3899340 A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways: +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | +---+---+---+---+ + +---+---+ +---+ +---+ +---+---+ + | | | | | | | | | | | | | | | | +---+---+ + +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | + +---+---+ + +---+---+ + +---+---+ + +---+---+ | | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ . +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | +---+ +---+ +---+---+---+---+ +---+---+---+---+ + +---+---+ | | | | | | | | | | | | | | | | | +---+---+---+---+ + +---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | + +---+---+ +---+ +---+ + + + + +---+---+ | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ . +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | | +---+---+ + +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | +---+---+---+---+ +---+ +---+ +---+---+---+---+ +---+---+ + | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ + + + + +---+---+ | | | | | | | | | | | | | | | + +---+---+ +---+ +---+ +---+---+---+---+ +---+---+---+---+ | | | | | | | | | | | | | | | | | | +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ . +---+---+---+---+ | | | | +---+ +---+ | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | +---+ +---+ | | | | +---+---+---+---+ The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37. CROSSREFS Main diagonal: A362143. Columns: A000012 (k = 0,1), A073028 (k = 2), A362144 (k = 3), A362145 (k = 4), A362146 (k = 5). Cf. A219924, A224697, A361216 (rectangular pieces). Sequence in context: A074749 A194524 A117136 * A139227 A269750 A292495 Adjacent sequences: A362139 A362140 A362141 * A362143 A362144 A362145 KEYWORD nonn,tabl AUTHOR Pontus von Brömssen, Apr 10 2023 STATUS approved

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Last modified December 11 02:45 EST 2023. Contains 367717 sequences. (Running on oeis4.)