why?

[Svante Linusson]

I think the prettiest way to explain this is to think of the "Fibonacci base".
One can just as writing every integer in decimal or binary base, write in
Fibonacci base.  Every integer can be represented as a string of zeros and
ones, example  1001010 will mean 21+5+2 and 1010101 will mean 21+8+3+1.  We
will have some identifications like 100 and 11 both representing 3, but if
we require that no two ones are consecutive  it is not difficult to see
that the representation of every positive integer is unique in the
Fibonacci base.

Wouter's question then becomes to ask for the smallest number that has
more than n ones in its Fibonacci base expression.
These numbers will clearly be (in Fibonacci base):
 101, 10101, 1010101, 101010101, 10101010101, ...
which are one less than
1000,100000,10000000,1000000000,100000000000, ...

combinacordially yours,


Svante Linusson


At 17.40 +0100 0-01-04, W. Meeussen wrote:
>the seq A027941 came up when I tried to find :
>a[n]=Least number not expressible as the sum
>of n Fibonacci numbers {1, 2, 3, 5, 8, 13, 21..
>
>Table[Position[CoefficientList[Series[(1+
>Sum[x^Fibonacci[i],{i,2,18}])^q,{x,0,Fibonacci[18]}],x],0,1,1]-1,{q,7}]
>
>{4, 12, 33, 88, 232, 609, 1596 ...
>
>
>ID Number: A027941
>Sequence: 1,4,12,33,88,232,609,1596,4180,10945,28656,75024,196417,
>514228,1346268,3524577,9227464,24157816,63245985,165580140,
>433494436,1134903169
>Name: Fibonacci(2n+1)-1.
>Comments: Also T(2n+1,n+1), T given by A027935
><http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.c
>gi?Anum=A027935>. Also first row of Inverse Stolarsky array. References C.
>Kimberling, "Interspersions and dispersions," Proceedings of the American
>Mathematical Society 117 (1993) 313-321.
>Links: ~See under Classic Sequences for more info
><http://www.research.att.com/njas/sequences/classic.html> ~Interspersions
><http://cedar.evansville.edu/ck6/integer/intersp.html>
>See also: Cf. A000045
><http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.c
>gi?Anum=A000045>, A035507
><http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.c
>gi?Anum=A035507>.
>Keywords: nonn,easy,nice
>Offset: 0
>Author(s): Clark Kimberling, ck6 at cedar.evansville.edu
>
>wouter.