cp's OEIS Frontend

A355242 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.

%I A355242 #11 Jul 05 2022 17:53:40
%S A355242 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 A355242 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 A355242 2,1,3,1,1,1,1,1,1,1,1,1,2,1,3,1,1,2,1,1,1,1,1
%N A355242 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 A355242 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 A355242 Hugo Pfoertner, <a href="/A355242/b355242.txt">Table of n, a(n) for n = 1..210</a>, rows 1..20 of triangle, flattened
%H A355242 Hugo Pfoertner, <a href="/A355242/a355242.pdf">(17 X 13)-rectangles with minimum number of covered grid points</a>
%e A355242 The triangle begins:
%e A355242     \ h 1  2  3  4  5  6  7  8  9 10 11 12 13
%e A355242    w \ --------------------------------------
%e A355242    1 |  1; |  |  |  |  |  |  |  |  |  |  |  |
%e A355242    2 |  1, 1; |  |  |  |  |  |  |  |  |  |  |
%e A355242    3 |  1, 1, 2; |  |  |  |  |  |  |  |  |  |
%e A355242    4 |  1, 1, 2, 1; |  |  |  |  |  |  |  |  |
%e A355242    5 |  1, 1, 2, 1, 3; |  |  |  |  |  |  |  |
%e A355242    6 |  1, 1, 2, 1, 3, 1; |  |  |  |  |  |  |
%e A355242    7 |  1, 1, 1, 1, 1, 1, 1; |  |  |  |  |  |
%e A355242    8 |  1, 1, 2, 1, 3, 1, 1, 2; |  |  |  |  |
%e A355242    9 |  1, 1, 1, 1, 3, 1, 1, 1, 1; |  |  |  |
%e A355242   10 |  1, 1, 2, 1, 3, 3, 1, 2, 1, 2; |  |  |
%e A355242   11 |  1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1; |  |
%e A355242   12 |  1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1; |
%e A355242   13 |  1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1
%Y A355242 Cf. A354702.
%Y A355242 A355241 is similar, but with slopes chosen from the list 1/2, 1, 3/2, 2, ... .
%K A355242 tabl,sign
%O A355242 1,6
%A A355242 _Hugo Pfoertner_, Jun 25 2022