A005897 (Points on surface of cube)

[all at abouthugo.de]

SeqFans,

can someone explain how to arrive at the 6*n^2+2
formula? There was a question on sci.math, how
many cubes are intersected by a sphere inscribed
in a cube, that is subdivided in n^3 smaller
cubes (see below)

Thanks
Hugo Pfoertner

<<
"HP" <yae9911 at netscape.net> wrote in message
news:89c716bb.0307150556.5626821d at posting.google.com...

> tmaier at optik.uni-erlangen.de (Tobias) wrote in message
news:<94040883.0307142226.24226261 at posting.google.com>...
> > Hallo all,
> >
> > I urgently need an answer to following:
> >
> > My task is to discretize (to sample) the surface of the unit sphere.
> > In terms of simplicity I've chosen to do this by deviding the
> > surrounding cubus (borderlenght=2) in N*N*N little cubi
> > (borderlenght=2/N). Only those little cubi that intersect the surface
> > of the unitsphere are of interest.
> >
> > My question now:
> >
> > How many little cubi will intersect the surface of the unitsphere in
> > dependency of N ?
[...]
> > Greetings
> > (mad) Tobias
>

My answer:

> The first few values are
>
> n  n_cut
> 1    1
> 2    8
> 3   26 (1 central cube of 27 not hit)
> 4   56 (8 central cubes of 64 not hit)
>
> Looking up [1,8,26,56] in the On-Line Encyclopedia of Integer Sequences
> gives
> http://www.research.att.com/projects/OEIS?Anum=A005897 :
> Points on surface of cube: 6n^2 + 2 (coordination sequence for b.c.c.
> lattice).
>
> The continuation of the sequence is:
> 1,8,26,56,98,152,218,296,386,488,602,728,866,1016,1178,1352,...

Message from Clive Tooth:

Ah... for n=6 that sequence gives 152. This is the mean of two answers
(176
and 128) that I gave in another post in this thread and corresponds to
viewing the cubi as neither open nor closed. I have no idea how the
expression 6n^2+2 arises, though.

-- 
Clive Tooth
http://www.clivetooth.dk
>>