[seqfan] Re: bases

[Martin Fuller]

More terms for your sequence (and for A008932 if it is the same):
1, 1, 2, 5, 17, 65, 292, 1434, 7875, 47098, 305226, 2122983, 15752080, 124015310, 1031857395

PARI:
A008932(n,pol=0)=
{
  local(a=0, i, pol2);
  !n & return(1);
  i = #pol;
  pol2 = pol^2;
  for (i=#pol, #pol2+1,
    a += A008932(n-1, pol+'x^i);
    !polcoeff(pol2,i) & break;
  );
  a
}

The largest value for each term in increasing bases is:
0, 1, 3, 5, 9, 13, 17, 21, 27, 33, 41, 47, 55
This is A123509 with a change in offset, and I agree with R. J. Mathar's conjecture that it is A001212(n)+1.

Martin Fuller

--- On Tue, 17/2/09, Robert G. Wilson, v <rgwv at rgwv.com> wrote:

> From: Robert G. Wilson, v <rgwv at rgwv.com>
> Subject: [seqfan] Re: bases
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Tuesday, 17 February, 2009, 6:50 PM
> 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.
> >
> >_______________________________________________
> >
> >Seqfan Mailing list - http://list.seqfan.eu/
> >
> >  
> >
> 
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> 
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