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A354704 T(w,h) is a lower bound for the maximum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, excluding the trivial case of an axis-parallel unshifted cover, where T(w,h) is a triangle read by rows.

%I A354704 #14 Dec 22 2024 10:51:23
%S A354704 2,3,5,5,8,13,6,10,15,18,8,12,20,24,32,9,14,23,27,36,41,10,17,25,30,
%T A354704 40,45,53,12,19,30,36,48,54,60,72,13,21,33,39,52,59,68,78,89,15,23,38,
%U A354704 45,60,68,75,90,98,113,16,25,40,48,64,72,81,96,105,120,128,17,28,43,52,68,77,88,102,114,128,137,149
%N A354704 T(w,h) is a lower bound for the maximum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, excluding the trivial case of an axis-parallel unshifted cover, where T(w,h) is a triangle read by rows.
%C A354704 Grid points must lie strictly within the covering rectangle, i.e., grid points on the perimeter of the rectangle are not allowed. See A354702 for more information.
%H A354704 Hugo Pfoertner, <a href="/A354704/b354704.txt">Table of n, a(n) for n = 1..210</a>, rows 1..20 of triangle, flattened
%H A354704 Hugo Pfoertner, <a href="/A354704/a354704.pdf">Illustrations of the initial terms up to T(5,5)</a>.
%e A354704 The triangle begins:
%e A354704     \ h  1   2   3   4   5   6   7    8    9   10   11   12
%e A354704    w \ ----------------------------------------------------
%e A354704    1 |   2;  |   |   |   |   |   |    |    |    |    |    |
%e A354704    2 |   3,  5;  |   |   |   |   |    |    |    |    |    |
%e A354704    3 |   5,  8, 13;  |   |   |   |    |    |    |    |    |
%e A354704    4 |   6, 10, 15, 18;  |   |   |    |    |    |    |    |
%e A354704    5 |   8, 12, 20, 24, 32;  |   |    |    |    |    |    |
%e A354704    6 |   9, 14, 23, 27, 36, 41;  |    |    |    |    |    |
%e A354704    7 |  10, 17, 25, 30, 40, 45, 53;   |    |    |    |    |
%e A354704    8 |  12, 19, 30, 36, 48, 54, 60,  72;   |    |    |    |
%e A354704    9 |  13, 21, 33, 39, 52, 59, 68,  78,  89;   |    |    |
%e A354704   10 |  15, 23, 38, 45, 60, 68, 75,  90,  98, 113;   |    |
%e A354704   11 |  16, 25, 40, 48, 64, 72, 81,  96, 105, 120, 128;   |
%e A354704   12 |  17, 28, 43, 52, 68, 77, 88, 102, 114, 128, 137, 149
%o A354704 (PARI) \\ See links in A354702 and A355244.
%Y A354704 Cf. A293330, A354702, A354705, A354706 (diagonal), A354707, A355244.
%Y A354704 Cf. A123690 (similar problem with circular disks).
%K A354704 nonn,tabl
%O A354704 1,1
%A A354704 _Hugo Pfoertner_, Jun 15 2022