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A355241
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T(w,h)/2 is the minimum slope >= 1/2 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 slope satisfying this condition exists.
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4
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1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 1, 6, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 6, 1, 2, 1, 2, 2, 1, 2, 6, 2, 2, 2, 2, 2, 2, 1, 1, 6, 6, 2, 1, 2, 1, 2, 2, 1, 2, 6, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2, 2
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OFFSET
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1,3
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COMMENTS
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No example of T(w,h) = -1 is known for w <= 20, i.e., the upper bound A354702(w,h) can always be achieved using a slope that is an integer multiple of 1/2. In the range w <= 20, T(17,13) = 3 is the only occurrence of the required slope 3/2.
For some rectangle dimensions it is possible to reach the value of A354702(w,h) with different slopes. In the simplest case, e.g., with the slopes 1/2 (T(w,h)=1) and 1 (A355242(w,h)=1). The linked file shows examples for some pairs of values (w,h) and the case of (10,10) with 3 different slopes.
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LINKS
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EXAMPLE
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The triangle begins:
\ h 1 2 3 4 5 6 7 8 9 10 11 12 13
w \ --------------------------------------
1 | 1; | | | | | | | | | | | |
2 | 1, 2; | | | | | | | | | | |
3 | 1, 1, 1; | | | | | | | | | |
4 | 2, 2, 1, 1; | | | | | | | | |
5 | 2, 2, 1, 1, 6; | | | | | | | |
6 | 2, 2, 1, 1, 6, 2; | | | | | | |
7 | 2, 2, 1, 2, 2, 2, 2; | | | | | |
8 | 2, 2, 1, 1, 6, 1, 2, 1; | | | | |
9 | 2, 2, 1, 2, 6, 2, 2, 2, 2; | | | |
10 | 2, 2, 1, 1, 6, 6, 2, 1, 2, 1; | | |
11 | 2, 2, 1, 2, 6, 2, 2, 1, 2, 1, 2; | |
12 | 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2; |
13 | 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2, 2
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PROG
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(PARI) /* see link. The program can be used to validate the given terms by calling it successively with the slope parameter k, starting with k = 1/2, 2/2=1, 3/2, (4/2 = 2 already covered by 1/2 via symmetry), 5/2, 6/2=3 for the desired rectangle size w X h , until the number of grid points given by A354702(w, k) is reached for the first time as a result. Without specifying the slope parameter, the program tries to approximate A354702(w, k) and determine a position of the rectangle maximizing the free space between peripheral grid points and the rectangle. */
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CROSSREFS
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A355244 is similar, but for maximizing the number of covered grid points.
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KEYWORD
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AUTHOR
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STATUS
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approved
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