[seqfan] A013583. Re: bloated sequences

[Richard Guy]

Without checking anything, let me rush in where
angels fear to tread.  One can get into trouble
with the Fib seq, 'cos 1 may be distinct from 1!

See Zeckendorf's theorem, for example.

Let's see what happens when we have two ones, say
a Blue one and a Red one.

1 = R = B,     2 = 2 = R+B,     3 = 3 = R+2 = B+2,

4 = R+3 = B+3 = R+B+2,       5 = 5 = 2+3 = R+B+3,

6 = R+5 = B+5,        7 = R+B+5 = R+B+2+3 = 2+5,

8 = 8 = R+2+5 = B+2+5 = 3+5,

9 = R+8 = B+8 = R+B+2+5 = R+3+5 = B+3+5,

10 = R+B+8 = R+B+3+5 = 2+8 = 2+3+5,

11 = R+2+8 = B+2+8 = R+2+3+5 = B+2+3+5 = 3+8,

12 = R+B+2+8 = R+3+8 = B+3+8 = R+B+2+3+5,

13 = 13 = R+B+3+8 = 2+3+8 = 5+8,

14 = R+13 = B+13 = R+2+3+8 = B+2+3+8 = R+5+8 = B+5+8,

etc (probably lots of mistakes by now!).

The numbers of equals signs are

for n = 1  2  3  4  5  6  7  8  9 10 11 12 13 14 ...
        2  2  3  3  3  2  3  4  5  4  5  4  4  6 ...
        ^     ^              ^  ^              ^
which suggests a sequence  1, 3, 8, 9, 14, ...

Should this be persoood ?    R.


On Wed, 18 Jun 2003, Antti Karttunen wrote:

> 
> 
> David Wilson wrote:
> 
> > Longest author line:        A013583 (202 chars)
> 
> Here is something which do not quite match:
> 
> ID Number: A013583
> Sequence:  1,3,8,16,24,37,58,63,97,105,152,160,168,249,257,270,406,401,
>            435,448,440,647,1011,673,723,715,1066,1058,1050,1092,1160,
>            1147,1694,1155,1710,1702,2647,1846,1765,1854,2736,1867,2757,
>            2744,2841,2990
> Name:      Smallest number that can be written as sum of distinct Fibonacci
>               numbers in n ways.
> See also:  Cf. A046815.
> Keywords:  nonn
> Offset:    1
> Author(s): Marjorie Bicknell-Johnson (marjohnson(AT)earthlink.net);
>               additional terms from Jeffrey Shallit
>               (shallit(AT)graceland.uwaterloo.ca); extended to 330 terms by
>               Daniel C. Fielder (dfielder(AT)ee.gatech.edu)
> Extension: The sequence continues:
>               2752,2854,2985,3019,4511,3032,6967,4456,3024,4477,4616,4451,7349,4629,
>            7218,4917,4621,4854,4904,7179,7166,4896,7200,7247,7310,7213,7831,8187,7488,
>            7205,11614,7480,7815,7857,7925,11593,18154,7912,11813,11682,11653,...
> 
> 
> ????
> 
> Yours,
> 
> Antti
> 
> PS. Using a subject like "Xtreme sequences" might trigger a few anti-spam filters.
>