# [seqfan] Re: bases

Robert G. Wilson, v rgwv at rgwv.com
Tue Feb 17 19:50:33 CET 2009

```Dear Sir,

Is this not just a rewording of sequence A008932
</%7Enjas/sequences/A008932>?

Bob.

David Newman wrote:

>I'd like someone to check some of my calculations before submitting them to
>the OEIS
>
>The idea for this sequence comes from a course in Additive Number Theory by
>Melvyn Nathanson.
>
>The set A of non-negative integers is called a basis if every
>non-negative integer can be written as the sum of two (not necessarily
>distinct) elements of A.
>
>Let's call a basis an increasing basis if its elements are arranged in
>increasing order, a0< a1< a2<...
>
>For example A126684 : 0, 1, 2, 4, 5, 8, 10, 16, 17, 20, 21, 32, 34, 40,...
>is an increasing basis.
>
>Next, consider the set of all initial subsequences of any length {a0, a1,
>a2,...,an} of all the increasing bases.  These can be ordered in the library
>ordering.  This sequence begins:
>
>0
>0, 1
>0, 1, 2
>0, 1, 3
>0, 1, 2, 3
>0, 1, 2, 4
>0, 1, 2, 5
>0, 1, 3, 4
>0, 1, 3, 5
>.
>.
>.
>How many such subsequences are there of length n?
>
>The numbers that I get, starting with a subsequence of length 1, are:
>
>
>1,1,2,5,17,65,292,1434
>
>I'd appreciate it if someone could check and extend this sequence.  When
>it's been checked I'll submit it through the site.
>
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>
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>
>
>

```