extending A005269

[Rob Pratt]

 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
> 
> 
> 
> 
>