[ October 22, 2001 ]
Have you ever... palindromes in the Golden String !
John McNamara (email)

The Golden String is a fascinating sequence of 0's and 1's that
is very closely related to the Fibonacci Numbers.
Other names for it are The Fibonacci Rabbit sequence (Dr Knott)
or The infinite Fibonacci word (N. Sloane).

The easiest algorithm to construct this Golden String
is in my opinion the following one :
Start with juxtaposition of 1 and 0 or 10 and set the pointer at the 0.
Now apply the next rule :

If digit pointed at is a 0 then append '1' else append '10'
Move the pointer one place to the right. Iterate.

We arrive thus at sequence 101 and already we moved
our pointer one digit further to the right (see underline).
Apply the rule again and we get 10110.
Continue and the Golden String gradually takes shape 1011010.

The first 100 'bits' of the string looks like this
10110101101101011010110110101101101011010110110101
10101101101011011010110101101101011011010110101101

two ways

Remember we started with 10 which is binary for 2 !
Now, John McNamara from Sydney Australia, discovered also
2 ways to see palindromes in this curious Golden String.

Firstly break the sequence up in real Fibonacci style :

1
0
11
010
11011
01011010
1101101011011
010110101101101011010
1101101011011010110101101101011011
0101101011011010110101101101011011010110101101101011010
etc.
All are palindromes with each string the length of the previous two,
formed by stringing together the previous two and changing the last
element from 0 to 1 or vice versa.
Well have you ever... amazing !

Secondly break the sequence up like this :

10
1
101
0110
1101011
01011011010
110110101101011011
01011010110110101101101011010
11011010110110101101011011010110101101101011011
etc.

all are palindromic and, with the exception of the artifice
at the start (which is in fact our start situation 10),
from the second line on contains the Lucas series numbers of bits.
1, 3, 4, 7, 11, 18, 29, ...
Have you ever... strikingly beautiful !

A065353
Decimal representation of palindromes
extracted from the Golden String using
ever increasing Fibonacci-style subdivisions.

A065354
Decimal representation of palindromes
extracted from the Golden String using
ever increasing Lucas-style subdivisions.

Website: mistermac
See also related WONplate 123