# [seqfan] bases

David Newman davidsnewman at gmail.com
Tue Feb 17 17:47:28 CET 2009

```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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