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%I #11 Jul 05 2022 17:53:40
%S 1,1,1,1,1,2,1,1,2,1,1,1,2,1,3,1,1,2,1,3,1,1,1,1,1,1,1,1,1,1,2,1,3,1,
%T 1,2,1,1,1,1,3,1,1,1,1,1,1,2,1,3,3,1,2,1,2,1,1,2,1,3,1,1,2,1,1,1,1,1,
%U 2,1,3,1,1,1,1,1,1,1,1,1,2,1,3,1,1,2,1,1,1,1,1
%N T(w,h) is the minimum integer slope >= 1 that can be chosen as orientation of a w X h rectangle such that the upper bound for the minimum number of covered grid points A354702(w,d) can be achieved by a suitable translation of the rectangle, where T(w,h) and A354702 are triangles read by rows. T(w,h) = -1 if no integer slope satisfying this condition exists.
%C T(17,13) = -1 is the first occurrence of the situation that it is not possible to reach the upper limit A354702(17,13) = 215 with a rectangle whose long side has an integer slope. (17 X 13)-rectangles with integer slope cannot cover less than 216 grid points. To achieve 215 grid points requires a slope of 3/2, i.e. A355241(17,13) = 3. See the linked file for related illustrations.
%H Hugo Pfoertner, <a href="/A355242/b355242.txt">Table of n, a(n) for n = 1..210</a>, rows 1..20 of triangle, flattened
%H Hugo Pfoertner, <a href="/A355242/a355242.pdf">(17 X 13)-rectangles with minimum number of covered grid points</a>
%e The triangle begins:
%e \ h 1 2 3 4 5 6 7 8 9 10 11 12 13
%e w \ --------------------------------------
%e 1 | 1; | | | | | | | | | | | |
%e 2 | 1, 1; | | | | | | | | | | |
%e 3 | 1, 1, 2; | | | | | | | | | |
%e 4 | 1, 1, 2, 1; | | | | | | | | |
%e 5 | 1, 1, 2, 1, 3; | | | | | | | |
%e 6 | 1, 1, 2, 1, 3, 1; | | | | | | |
%e 7 | 1, 1, 1, 1, 1, 1, 1; | | | | | |
%e 8 | 1, 1, 2, 1, 3, 1, 1, 2; | | | | |
%e 9 | 1, 1, 1, 1, 3, 1, 1, 1, 1; | | | |
%e 10 | 1, 1, 2, 1, 3, 3, 1, 2, 1, 2; | | |
%e 11 | 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1; | |
%e 12 | 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1; |
%e 13 | 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1
%Y Cf. A354702.
%Y A355241 is similar, but with slopes chosen from the list 1/2, 1, 3/2, 2, ... .
%K tabl,sign
%O 1,6
%A _Hugo Pfoertner_, Jun 25 2022