[seqfan] Re: 0-additive and first differences

Heinz, Alois alois.heinz at hs-heilbronn.de
Thu May 14 01:09:19 CEST 2015

In a 0-additive sequence each term is the sum of two distinct
*earlier* terms in exactly 0 ways.

If you want the lexicographically first 0-additive sequence such
that the terms in the sequence and their absolute first differences
are all distinct you get:

1, 3, 7, 12, 18, 26, 9,  20, 34, 24, 39, 55, 22, 45, 66, 28, ...
. 2, 4, 5, 6,  8,  17, 11, 14, 10, 15, 16, 33, 23, 21, 38, ...

But then some positive numbers do not occur.

Please note that 12 is the sum of 3 and 9 but 9 is not an
earlier term.

If you want the lexicographically first sequence such that the terms
in the sequence and their absolute first differences are all distinct
and each term is the sum of two distinct *other* terms in exactly 0 ways
you get:

1, 3, 7, 12, 18, 26, 16, 31, 20, 37, 50, 22, 41, 64, 35, 56, 83, ...
. 2, 4, 5, 6,  8,  10, 15, 11, 17, 13, 28, 19, 23, 29, 21, 27, ...

But then again some positive numbers do not occur.

Best regards, Alois

Am 13.05.2015 um 22:05 schrieb Frank Adams-Watters:
> OK. 0-additive is different from what I thought.
>
> So, I want the lexicographically first sequence such that the sequence and its absolute first differences are all distinct. This starts as I stated, and it isn't in the database. A095115 is an approximation to this, but it doesn't really do any backtracking and is thus finite.
>