A389352 - OEIS

A389352

Manhattan distance to the origin of point n while traversing the half plane by integer points in rectangular layers starting from n=1 at the origin.

0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Jens Ahlström, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = |A389649(n)| + A389648(n).

EXAMPLE

Path begins:

25--24--23--22--21--20--19 Y=3

| |

26 9--10--11--12--13 18 Y=2

| | | |

27 8 5---4---3 14 17 Y=1

| | | | | |

28 7---6 1---2 15--16 Y=0

----------------------------

X = -3 -2 -1 0 1 2 3

For n=8, the path has come to the point (-2, 1), which has the Manhattan distance a(8) = abs(-2) + abs(1) = 3.

PROG

(Python)

def traverse_upper_halfplane(n):

points = [(0, 0)]

arch = 1

sign = 1

while len(points) < n:

for y in range(arch+1):

points.append((sign*arch, y))

for x in range(arch-1, -arch-1, -1):

points.append((sign*x, arch))

for y in range(arch-1, -1, -1):

points.append((-sign*arch, y))

arch += 1

sign = - sign

return points[:n]

l = [abs(x)+y for (x, y) in traverse_upper_halfplane(91)]

print(l)

CROSSREFS

Cf. A213088 (quadrant), A214526 (whole plane), A389648 (Y), A389649 (X).

Sequence in context: A317586 A303780 A234022 * A261273 A097454 A139803

Adjacent sequences: A389349 A389350 A389351 * A389353 A389354 A389355

KEYWORD

nonn,easy

AUTHOR

Jens Ahlström, Oct 01 2025

STATUS

approved