# [seqfan] Walk like an Egyptian

Jens Voß jens at voss-ahrensburg.de
Fri Feb 20 20:00:42 CET 2009

```Hello Sequence lovers,

Recently I was dealing a little with certain sums of egyptian fractions
which lead me to exploring the different ways of splitting an integer
into a sum of decreasing (though not necessarily strictly decreasing)
egyptian fractions of given length n. To my surprise, this sequence did
not appear to be in the OEIS, so I calculated the first few terms.

In order to structure the different sums, I grouped them into sequences
adding up to an integer k (which of course then has be be in {1, .., n})
and obtained the following pyramid:

_
__ -
/     __   \
/   _ -    |
| '  | (_)  |
|  __  /   /          1
\ \  __  /          1     1
-            3     1     1              a'!   _,,_ a'!   _,,_
14     4     1     1             \\_/    \  \\_/    \
147    17     4     1     1           \, /-( /'-,\, /-( /'-,
_.,-*~ 3462   164    18     4     1     1 .,-*~'^'//\ //\\   //\ //\\
|                                                                     *,
|                                                                      |
`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'

(submitted to the OEIS as A156869; the sum over all k is A156871)

Interestingly, all the lines going down to the right seem to converge,
and as fas as I was able to tell from my small set of sample data, the
point at which those line becomes constant appears to lie on the column
immediately to the left of the central axis (I could have made the
offset 0, in which case an additional diagonal line with entries
1, 0, 0, 0, ... would have to be added to the left side of the shape -
then the presumable "points of convergence" would in fact be the central
axis).

I stated this observation as a conjecture in A156869, however I have
only very little empirical evidence of this hypothesis, much less a
proof to it. (The sequence 1, 4, 18, 168, 3648, ... in the column are
submitted as A156870.)

Are there any egyptologists on this mailing list that can either come up
with additional terms to the sequences A156869 - A156871 or prove the
conjecture?

Best regards,
Jens

```