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 A362258 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, up to rotations and reflections, 0 <= k <= n. 5
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 1, 1, 2, 4, 13, 20, 1, 1, 4, 8, 33, 125, 277, 1, 1, 6, 12, 72, 403, 2505, 7855, 1, 1, 9, 22, 204, 1438, 12069, 101587, 487662 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 LINKS Table of n, a(n) for n=0..44. FORMULA T(n,k) >= A362142(n,k)/4 if n != k. T(n,n) >= A362142(n,n)/8. 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 1 1 4 | 1 1 2 2 4 5 | 1 1 2 4 13 20 6 | 1 1 4 8 33 125 277 7 | 1 1 6 12 72 403 2505 7855 8 | 1 1 9 22 204 1438 12069 101587 487662 See A362142 for an illustration of T(5,4) = 13. The following table shows which sets of squares can tile the n X k rectangle in T(n,k) ways. A list x_1, ..., x_j represents a set of x_1 squares of side 1, ..., x_j squares of side j. When there are multiple solutions they are shown on separate lines. For (n,k) = (4,3), for example, the maximum number T(4,3) = 2 of tilings is obtained both for the set of 8 squares of side 1 and 1 square of side 2, and for the set of 4 squares of side 1 and 2 squares of side 2. n\k| 1 2 3 4 5 6 7 8 ---+------------------------------------------------ 1 | 1 2 | 2 4 | 0,1 3 | 3 6 9 | 2,1 5,1 | 0,0,1 4 | 4 4,1 8,1 8,2 | 4,2 5 | 5 6,1 7,2 12,2 13,3 | 2,2 6 | 6 4,2 10,2 12,3 14,4 20,4 7 | 7 6,2 13,2 12,4 19,4 22,5 25,6 8 | 8 8,2 12,3 16,4 20,5 24,6 23,6,1 27,7,1 CROSSREFS Main diagonal: A362259. Columns: A000012 (k = 0,1), A362260 (k = 2), A362261 (k = 3), A362262 (k = 4), A362263 (k = 5). Cf. A227690, A361221 (rectangular pieces), A362142. Sequence in context: A282627 A004565 A068449 * A068450 A071436 A214741 Adjacent sequences: A362255 A362256 A362257 * A362259 A362260 A362261 KEYWORD nonn,tabl,more AUTHOR Pontus von Brömssen, Apr 15 2023 STATUS approved

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Last modified August 11 11:53 EDT 2024. Contains 375069 sequences. (Running on oeis4.)