cp's OEIS Frontend

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.

%I A362258 #10 Apr 16 2023 08:37:54
%S A362258 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,
%T A362258 12,72,403,2505,7855,1,1,9,22,204,1438,12069,101587,487662
%N 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.
%F A362258 T(n,k) >= A362142(n,k)/4 if n != k.
%F A362258 T(n,n) >= A362142(n,n)/8.
%e A362258 Triangle begins:
%e A362258   n\k| 0  1  2  3   4    5     6      7      8
%e A362258   ---+----------------------------------------
%e A362258   0  | 1
%e A362258   1  | 1  1
%e A362258   2  | 1  1  1
%e A362258   3  | 1  1  1  1
%e A362258   4  | 1  1  2  2   4
%e A362258   5  | 1  1  2  4  13   20
%e A362258   6  | 1  1  4  8  33  125   277
%e A362258   7  | 1  1  6 12  72  403  2505   7855
%e A362258   8  | 1  1  9 22 204 1438 12069 101587 487662
%e A362258 See A362142 for an illustration of T(5,4) = 13.
%e A362258 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.
%e A362258   n\k| 1   2      3     4     5     6      7       8
%e A362258   ---+------------------------------------------------
%e A362258   1  | 1
%e A362258   2  | 2  4
%e A362258      |    0,1
%e A362258   3  | 3  6     9
%e A362258      |    2,1   5,1
%e A362258      |          0,0,1
%e A362258   4  | 4  4,1   8,1    8,2
%e A362258      |          4,2
%e A362258   5  | 5  6,1   7,2   12,2  13,3
%e A362258      |    2,2
%e A362258   6  | 6  4,2  10,2   12,3  14,4  20,4
%e A362258   7  | 7  6,2  13,2   12,4  19,4  22,5  25,6
%e A362258   8  | 8  8,2  12,3   16,4  20,5  24,6  23,6,1  27,7,1
%Y A362258 Main diagonal: A362259.
%Y A362258 Columns: A000012 (k = 0,1), A362260 (k = 2), A362261 (k = 3), A362262 (k = 4), A362263 (k = 5).
%Y A362258 Cf. A227690, A361221 (rectangular pieces), A362142.
%K A362258 nonn,tabl,more
%O A362258 0,13
%A A362258 _Pontus von Brömssen_, Apr 15 2023