OFFSET
1,3
COMMENTS
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.
LINKS
Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened
Hugo Pfoertner, Different slopes with the same number of grid points covered.
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, 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
PROG
(PARI) /* See Pfoertner 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. */
CROSSREFS
KEYWORD
AUTHOR
Hugo Pfoertner, Jun 27 2022
STATUS
approved