WON plate118 | World!OfNumbers [ October 22, 2001 ] Have you ever... palindromes in the Golden String ! John McNamara (email)  What is the Golden String ?  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 movedour 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 palindromesextracted from the Golden String usingever increasing Fibonacci-style subdivisions. A065354 Decimal representation of palindromesextracted from the Golden String usingever increasing Lucas-style subdivisions. Website: mistermac See also related WONplate 123 A000118 Prime Curios! Prime Puzzle Wikipedia 118 Le nombre 118
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