A005897 (Points on surface of cube)

[all at abouthugo.de]

> Neil, SeqFans,
> 
> it was not obvious to me that the number of
> sub-cubes cut (not just touched in an edge or
> face) is equal to the number of lattice points
> on the surface of the subdivided cube. If this
> is true, than it should be added as a comment
> to A005897. Or is there a new sequence?

With the help of a little program I was able
to disprove my initial guess: The numbers of
cut spheres starts to deviate from A005897 for
higher n.
A005897:
1,8,26,56,98,152,218,296,386,488,602,728,866,1016,
1178,1352,
Cut Cubes:
  8,26,56,98,152,194,272,362,440,530,656,746,872,
1034,1160,...

The program and the results up to n=100 are at
http://www.randomwalk.de/sequences/cutcub.txt
(see comment for counting of touched spheres)

My results are similar to those posted by
Dave Seaman to sci.math NG:

Here is a copy of his contribution:
<<
I suppose one could view each of the "cubi" as an n-dimensional interval
that is a Cartesian product of half-open intervals, so that they exactly
partition the space.  This would lead to the half-way answer, I think.

I wrote a lisp program to compute the results up to n = 10 and I seem to
be in agreement with most of the results that have been posted, but
there
are some differences compared to the A005897 sequence for n > 5:

        n       count
        -       -----
        1       1
        2       8
        3       26
        4       56
        5       98
        6       176 or 128
        7       194
        8       272
        9       362
        10      464 or 416

I also thought of looking up the sequence, but my sequence gave no
matches.

-- 
Dave Seaman

>>

Hugo Pfoertner