[seqfan] how to scan a 3D array

Chris Starling chaosorder4 at gmail.com
Mon Dec 7 16:13:26 CET 2015


Hello, sequence people...

Longtime lurker here.

I have quite a few 3D arrays of integers, based on quantities for
trivariate figurate forms.  With these, typically, there are three sets of
edges on a figurate form, each of which have a unit count.  When those
three counts are plugged into a formula, a total units quantity for that
figurate form is found, and that can go into the sequence.

But the question is, where in the sequence?

I have seen sequences that are  2D arrays, and in the description is,
"triangle read by rows", which is understandable enough.  But there are at
least two ways of doing that.  For the same 2D array, one could repeatedly
scan from UR to LL diagonally, or from LL to UR.  For a third possibility,
one could alternate, so that there is always array adjacency preserved
between any two neighbors in the sequence.  And it gets expectably more
complex with 3D.  There are at least six ways to do carriage-return linear
scans, before considering adjacency and spirals.

So what I'm wondering is, is there some convention, based on the
distinctions between variables, or should I just pick one way, and label
how I scanned it?

An example array starts as follows:

    1

    6    4
  13

  19  16  10
  38  28
  55

  44  40  31  20
  79  68  50
116  92
147

The UL corner of each triangle is an octahedron, the UR is a tetrahedron,
and the bottom corner is a cuboctahedron.  These are quantities of spheres
packed in FCC.  The rest are in the field of varying truncation between
those, that grows with each layer.

Just like with 2D arrays, there are multiple smaller order sequences
automatically included within it.

Any help or guidance will be much appreciated.
Sincerely, Chris.


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