extending A005269

[Emeric Deutsch]

Sorry. My fault. I neglected to mention that the sequence 
b_j is nondecreasing. My Maple program does take care of that.
Thanks Rob.
Emeric
P.S. I will suggest to Neil to give the full definition of these 
sequences.


On Fri, 4 Feb 2005, Rob Pratt wrote:

> From the counts, it appears that the definition also requires b_{j-1} <= b_j.  Could someone please update the OEIS entry with the full definition?  Currently, it has none.
> 
> Rob
> 
> > -----Original Message-----
> > From: Emeric Deutsch [mailto:deutsch at duke.poly.edu] 
> > Sent: Friday, February 04, 2005 11:20 AM
> > To: seqfan at ext.jussieu.fr
> > Subject: extending A005269
> > 
> > Dear seqfans,
> > a(n) is the number of sequences b_1,b_2,...b_n of length n 
> > such that b_1=b_2=1 and b_j <= b_{j-2}+b_{j-1} for all j>2.
> > They are called sub-Fibonacci sequences.
> > The A005269 entry in OEIS is given below.
> > With a Maple program I get very fast the first 10 terms of 
> > the sequence, relatively fast the 11th term 1067599 but the 
> > next term eludes me even after several hours of computer time.
> > I am not familiar with other programs like Mathematica, PARI, 
> > etc. I wonder if somebody is interested to work on this.
> > 
> > For the purpose of counting, it is sufficient to construct 
> > only the pair of the last two entries of a sub-Fibonacci 
> > sequence. Thus,
> > F_2 = [1,1];
> > F_3 = [1,1],[1,2];
> > F_4 = [1,1],[1,2],[2,2],[2,3];
> > F_5 = [1,1],[1,2],[2,2],[2,3],[2,2],[2,3],[2,4],[3,3],[3,4],[3,5];
> > (incidentally, offset should be 2 and first term in the 
> > present sequence should be dropped).
> > F_j contains the "mutilated" sub-Fibonacci sequences of length j.
> > Obviously, a pair can occur several times in F_j.
> > A pair p=[a,b] in F_j generates the following pairs in F_{j+1}:
> > [b,b],[b,b+1],...,[b,b+a].
> > For this, I am using in Maple the function 
> > f:=p->seq([p[2],p[2]+i],i=0..p[1]);
> > Then:
> > > F[2]:=[1,1]; a[2]:=nops([F[2]]); # initialization
> > yielding a[2]=1 and then
> > > for n from 3 to 10 do F[n]:=seq(f([F[n-1]][i]),i=1..a[n-1]):
> > a[n]:=nops([F[n]]): print(a[n]) od:
> > yielding very fast 2,4,10,31,127,711,5621,64049.
> > 
> > Emeric
> > 
> > ID Number: A005269 (Formerly M1234)
> > URL:       http://www.research.att.com/projects/OEIS?Anum=A005269
> > Sequence:  1,1,2,4,10,31,127,711,5621,64049,1067599
> > Name:      Number of sub-Fibonacci sequences of length n.
> > References Fishburn, Peter C.; Roberts, Fred S.; Elementary sequences,
> >            sub-Fibonacci sequences. Discrete Appl. Math. 44 
> > (1993), no.
> > 1-3, 261-281.
> > See also:  Adjacent sequences: A005266 A005267 A005268 
> > this_sequence A005270 A005271 A005272
> >            Sequence in context: A001647 A007177 A005268 
> > this_sequence A070900 A071954 A000736
> > Keywords:  nonn
> > Offset:    1
> > Author(s): njas
> > 
> > 
> > 
> > 
> > 
> 
>