A290190 Minimum sum of mutual Manhattan distances of n distinct grid points in a cubic lattice.

1, 4, 8, 16, 25, 36, 48, 68, 89, 112, 136, 168, 201, 238, 278, 322, 369, 420, 472, 536, 599, 664, 734, 812, 891, 972, 1062, 1156, 1249, 1344, 1440, 1560, 1681, 1806, 1933, 2064, 2199, 2340, 2480, 2632, 2785, 2940, 3096, 3268, 3443, 3624, 3808, 4008, 4209, 4416, 4625

Offset

2,2

Comments

Sum of mutual L1-distances of locations in a 3-dimensional n-town of optimum shape.

Conjecture: a(k^3) = A292045(k), i.e., fully populated cubes are optimal 3-dimensional n-towns. - Hugo Pfoertner, Jan 08 2026

The conjecture is false. Optimal 3-dimensional k^3-towns exist for k >= 4 with distance sums < A292045(k). For example, there are 64-towns (see linked illustration) with distance sum = 7539 < A292045(4) = 7680 and 125-towns with distance sum 36510 < A292045(5) = 37500. - Hugo Pfoertner, Jan 26 2026

Links

Hugo Pfoertner, Table of n, a(n) for n = 2..515

Hugo Pfoertner, A 64-town with distance sum 7539, (2026).

Hugo Pfoertner, A 125-town with distance sum 36510, (2026).

Hugo Pfoertner, A 216-town with distance sum 131708, (2026).

Hugo Pfoertner, Graph of the deviation of A290190(n) to the fitted function 0.4565*n^2.3393 for n<=800, (2026).

Hugo Pfoertner, Examples of some conjecturally optimal 3D n-towns for n>1300, (2026).

Hugo Pfoertner, Examples of optimal point configurations for n = 2...48, 60, 81, 2026-03-07.

Example

a(2)=1: Grid points (1 2 2),(1 1 2)
a(3)=4: (1 1 1),(1 2 1),(2 1 1)
a(4)=8: (1 1 1),(2 1 1),(1 2 1),(2 2 1)
a(5)=16: (1 2 2),(2 2 1),(2 2 2),(1 1 1),(1 2 1)
a(6)=25: (2 1 2),(1 1 1),(2 2 1),(2 1 1),(1 2 1),(2 2 2)
a(10)=89: (2 2 2),(1 3 1),(1 1 1),(1 2 2),(2 3 2),(2 1 1),(1 3 2),(1 2 1),(2 2 1),(2 3 1)

Crossrefs

Cf. A193452, A292045.

A393008 contains a link to a comprehensive comparison of spherical configurations to the optimal towns of the present sequence.

Sequence in context: A389210 A369566 A022560 * A193452 A003451 A277029

Adjacent sequences: A290187 A290188 A290189 * A290191 A290192 A290193

Keyword

nonn

Author

Hugo Pfoertner, Jul 23 2017

Extensions

More terms from Hugo Pfoertner, Jan 26 2026

Status

approved