Basic Examples

The first number whose representation takes three summands is 12:

○
ZeckendorfRepresentation
[12]
In[]:=
{1,0,1,0,1}
Out[]=

This corresponds to 8 + 3 + 1.

Reverse@Fibonacci[{2,4,6}]
In[]:=
{8,3,1}
Out[]=

The first number whose representation takes four summands is 33:

○
ZeckendorfRepresentation
[33]
In[]:=
{1,0,1,0,1,0,1}
Out[]=
Withz=
○
ZeckendorfRepresentation
[33],Prepend[Reverse@z,0].Fibonacci@Range[1+Length@z]
In[]:=
33
Out[]=

Neat Examples

Fibonacci@6
In[]:=
8
Out[]=

There are
F
k
Zeckendorf representations of length
k
; for example, here are the 13 representations of length 7:

○
ZeckendorfRepresentation
/@Range[21,33]//Grid
In[]:=
1
0
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
0
1
0
0
0
1
0
0
1
0
0
0
1
0
1
1
0
0
1
0
0
0
1
0
0
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
0
0
0
1
0
1
0
0
0
1
1
0
1
0
0
1
0
1
0
1
0
1
0
0
1
0
1
0
1
0
1
Out[]=

This visualizes the same pattern:

Framed@Graphics@Raster@Reverse1-
○
ZeckendorfRepresentation
/@Range[21,33]
In[]:=
Out[]=