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A355244
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 lower bound for the maximum number of covered grid points A354704(w,d) can be achieved by a suitable translation of the rectangle, where T(w,h) and A354704 are triangles read by rows. T(w,h) = -1 if no slope satisfying this condition exists.
3
1, 1, 1, 1, 2, 2, 1, 3, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 6, 2, 2, 2, 1, 6, 2, 6, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2, 1, 3, 2, -1, 2, 2, 3, 2, 1, 2, -1, 3, 2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2, 1, 2
OFFSET
1,5
LINKS
Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened
Hugo Pfoertner, PARI program
EXAMPLE
The triangle begins:
\ h 1 2 3 4 5 6 7 8 9 10 11 12 13
w \ --------------------------------------
1 | 1; | | | | | | | | | | | |
2 | 1, 1; | | | | | | | | | | |
3 | 1, 2, 2; | | | | | | | | | |
4 | 1, 3, 2, 2; | | | | | | | | |
5 | 1, 1, 2, 2, 2; | | | | | | | |
6 | 1, 1, 2, 2, 2, 2; | | | | | | |
7 | 1, 6, 2, 2, 2, 1, 6; | | | | | |
8 | 2, 6, 2, 2, 2, 2, 2, 2; | | | | |
9 | 1, 1, 2, 2, 2, 1, 1, 2, 1; | | | |
10 | 2, 1, 2, 2, 2, 2, 2, 2, 2, 2; | | |
11 | 2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2; | |
12 | 1, 3, 2,-1, 2, 2, 3, 2, 1, 2,-1, 3; |
13 | 2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2, 1, 2
.
The first linked illustration shows examples where 2 slopes lead to the same number of covered grid points, where then the smallest multiple of 1/2 is used as a term in the sequence.
The second illustration shows the two examples where it is not possible to cover the maximum number of grid points with a rectangle whose side slope is an integer multiple of 1/2.
PROG
(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 A354704(w, k) is reached for the first time as a result. If the slope parameter is not specified, the program attempts to approximate A354704(w, k) and determine a location of the rectangle that maximizes the free margin between the peripheral grid points and the perimeter of the rectangle. */
CROSSREFS
Cf. A355241 (similar, but with number of covered grid points minimized).
Sequence in context: A124018 A309288 A235061 * A364730 A283004 A111709
KEYWORD
tabl,sign
AUTHOR
Hugo Pfoertner, Jun 29 2022
STATUS
approved