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A351042 Minimal number of steps for a Racetrack car (using von Neumann neighborhood) to go around a circle of radius n. 8
9, 12, 13, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 32, 34, 34, 36, 36, 37 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The car moves according to the rules of the game of Racetrack with von Neumann neighborhood, i.e., if P, Q, and R are three successive positions of the car, one coordinate of the second difference (acceleration vector) P - 2Q + R must be 0, and the other 1, 0, or -1. The car starts with zero velocity at a point (x,0) for some integer x >= n, and finishes when it passes, or lands on, the positive x-axis after a complete counterclockwise lap around the origin. The line segments between successive positions must be outside or on the circle with center in (0,0) and radius n.
LINKS
Wikipedia, Racetrack
FORMULA
a(n) = min {k >= 8; A351351(k)/A351352(k) >= n^2}.
a(n) >= A351041(n).
EXAMPLE
The following diagrams show examples of optimal trajectories for n = 1, 2, 3. The origin is marked with an asterisk.
.
a(1) = 9:
. 3 2 . .
4 . . 1 .
5 . * 0 9
. 6 7 8 .
.
a(2) = 12:
. 4 3 2 . .
5 . . . 1 .
6 . * . 0 12
7 . . . 11 .
. 8 9 10 . .
.
a(3) = 13:
. . . 4 . 3 . . . .
. 5 . . . . . 2 . .
6 . . . . . . . 1 .
7 . . . * . . . 0 13
8 . . . . . . . . .
. 9 . . . . . 12 . .
. . . 10 . 11 . . . .
CROSSREFS
Sequence in context: A335168 A295486 A032687 * A259313 A170951 A044859
KEYWORD
nonn,more
AUTHOR
STATUS
approved

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Last modified June 13 22:21 EDT 2024. Contains 373391 sequences. (Running on oeis4.)