# A054770, again.

karttu at megabaud.fi karttu at megabaud.fi
Fri Jun 9 02:36:51 CEST 2000

```John Conway wrote:
>
>    This brings to mind Kimberling's wonderful and little-known theory
> about the entire set of Fibonacci-like sequences, which is much more
> interesting than discussing where they should properly start.
>
>    So let me describe this theory (in my own language, rather than
> Kimberling's).  It leads to a table:
>
>         -1  0| 1  2  3  4  5  6  7  8  9 10 ...
>          ----+---------------------------------
>          0  1| 1  2  3  5  8 13 21 34 55 89 ...
>          1  3| 4  7 11 18 29 47 76 ...
>          2  4| 6 10 16 26 42 68 ...
>          3  6| 9 15 24 39 63 ...
>          4  8|12 20 32 52 84 ...
>          5  9|14 23 37 60 97 ...
>          6 11|17 28 45 73 ...
>          7 12|19 31 50 81 ...
>          8 14|22 36 58 94 ...
>          9 16|25 41 66 ...
>         10 17|27 44 71 ...
>          ...........
>

Dear John and Dear Jud,

this is just the Wythoff Array, A035513, about which
Neil gives a detailed explanation at
http://www.research.att.com/~njas/sequences/classic.html

And it was just the related sequences like
A000201 Lower Wythoff sequence (a Beatty sequence): [ n*tau ].
(the column to the left of the broken line in that array,
1,3,4,6,8,9,11,12,14,16,...)

and
A022342 Integers with "even" Zeckendorf expansions
or Fibonacci successor to n-1.
a(n) = [ n tau^2 ] - n - 1; or [ n tau ] -1.

which clearly seem to be related to the formula

a_n = [((5+sqrt(5))/2)n]-1
(= [(phi + 2)*n] - 1, where phi is the golden ratio (1+sqrt(5))/2)
(= tau in the other references).

as conjectured for A054770 by David W. Wilson, that I immediately
suggested to Antreas, that if he added the "missing" 2 to his set
of Lucas numbers, then they probably would make a complete sequence
(= allowing representation of all the natural numbers as sums
of distinct Lucas numbers) (for now, this is just my hypothesis
that 2 would make it),
and then find some way to convert sums of distinct Lucas numbers
to sums of distinct, non-consecutive Fibonacci numbers,
(e.g. by using the identity L(n) = F(n-1)+F(n+1) and other tricks)
i.e. Zeckendorf Expansions, and then see how that "2", if present,
affects the pattern of the produced Zeckendorf Expansion. And what kind of
Zeckendorf Expansions are produced by that conjectured formula,
and whether the patterns are same.

(Because if the Lucas sequence is a complete sequence with that L(0) = 2
present, then an integer is in A054770 if and only if
its representation as a sum of distinct Lucas numbers requires
that 2. Simple.)

Terveisin,

Antti Karttunen

```