A306656 Number of ways to fill a 3D matrix with n distinct values.

1, 1, 6, 18, 144, 360, 6480, 15120, 403200, 2177280, 32659200, 119750400, 8622028800, 18681062400, 784604620800, 11769069312000, 313841848320000, 1067062284288000, 115242726703104000, 364935301226496000, 43792236147179520000, 459818479545384960000

Offset

0,3

Comments

This sequence is a generalization of A323295 to the 3D case. Usually, in multidimensional data related applications (i.e., images, MRI), data is vectorized and then processed. However, because of vectorization, the spatial information in the data is lost. This reverse mapping shows the possible number of spatial states the original data could have been in.

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = A007425(n) * n! for n > 0, a(0) = 1.

Example

For n = 6, a(6) = 6480, A007425(6) = 9 namely there are 9 ways to arrange 6 voxels into a 3D matrix: [1,1,6], [1,6,1], [6,1,1], [2,3,1], [3,2,1], [2,1,3], [3,1,2], [1,2,3], [1,3,2]. Then there are 6! ways to fill it with the numbers. 9*6! = 6480.

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(tau(d), d=divisors(n))*n!):
seq(a(n), n=0..23);  # Alois P. Heinz, Mar 03 2019

Crossrefs

Cf. A007425, A077592, A323295.

Sequence in context: A222913 A091054 A012774 * A027744 A196868 A077531

Adjacent sequences: A306653 A306654 A306655 * A306657 A306658 A306659

Keyword

nonn

Author

Yigit Oktar, Mar 03 2019

Extensions

More terms from Alois P. Heinz, Mar 03 2019

Status

approved