[seqfan] Re: Walk like an Egyptian

franktaw at netscape.net franktaw at netscape.net
Fri Feb 20 22:20:16 CET 2009

Yes, the diagonals are constant for n <= 2k.  Any such sequence must 
have at least one 1; remove that 1, and you get a sequence for n-1,k-1.

Franklin T. Adams-Watters

-----Original Message-----
From: Jens Voß <jens at voss-ahrensburg.de>

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.


(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

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

Best regards,


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