[seqfan] A013583. Re: bloated sequences

[David Wilson]

Starting with the original problem:

The number of ways to add up to n with distinct Fibonacci
numbers is

A000119 = (1 1 1 2 1 2 2 1 3 2 2 3 1 3 3 2 4 2 3 3 1...)

indexed starting at 0.  This is A000119.

The least inverse of A000119 is

A013583 = (0 3 8 16 24 37 58 63 97 105 152 160 168 249 257 270...)

indexed starting at 1.

I feel that A013583(1) should be changed from 1 to 0.  Also:

%F A013583 A000119(A013583(n)) = n

At any rate, if we allow the extra 1 among the Fibonaccis,
then the number of ways to add up to n with nondistinct
Fibonaccis is

A000121 = (1 2 2 3 3 3 4 3 4 5 4 5 4 4 6 5 6 6 5 6 4...)

%F A000121 a(n) = A000119(n) + A000119(n-1).

In your note, you had A000121(6) = 2, whereas actually A000121(6) = 4.
You gave the two sums

 6 = R+5 = B+5

whereas there are four sums:

  6 = R+5 = B+5 = R+2+3 = B+2+3.

The sequence you were really trying to get at was the least inverse of
A000121, which is not in the OEIS

%I A000000
%S A000000 0,1,3,6,9,14,22,24,37,40,58,61,64,95,98,103,155,153,166,171,168,247,386
%T A000000 257,276,273,407,404,401,417,443,438,647,441,653,650,1011,705,674,708,1045
%U A000000 713,1053,1048,1085,1142,1051,1090,1140,1153,1723,1158,2661,1702,1155,1710
%N A000000 Smallest number that can be written as a sum of Fibonacci numbers in n ways, counting 1 twice as Fibonacci number.
%O A000000 1,3
%F A000000 A000121(A000000(n)) = n.
%A A000000 rkg


----- Original Message ----- 
From: Richard Guy 
To: Antti Karttunen 
Cc: David Wilson ; Sequence Fanatics 
Sent: Wednesday, June 18, 2003 5:52 PM
Subject: Re: [seqfan] A013583. Re: bloated sequences


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.
> 
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