Two observations (i) we can replace 110 with 1 and 10 with 0 in the original sequence and get a reduced sequence and it seems we can reduce the resulting sequence in a similar manner. so it forms a kind of self similar sequence... (ii) 55 and 89 are Fibonacci numbers Christoph ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of Robert Munafo [mrob27@gmail.com] Sent: Saturday, June 23, 2012 8:31 PM To: math-fun Cc: Mike Stay Subject: Re: [math-fun] The final Google Turing puzzle: what is 11010110110101101011011... On 6/23/12, Mike Stay <metaweta@gmail.com> wrote:
That has period 90. It repeats this binary string: 11010110110101101011011010110110101101011011010110101101101011011010110101101101011011010
Not quite. Your string (which has 89 characters, making more sense given your conclusion) occurs twice, then we get 1101011011010110101101101011011010110101101101011011010 which has 55 characters, then another of the 89-character one. If we set A to be your 89-character string, and B = the 55-character string, then the output of the mystery Turing machine starts: A A B A A B A B A A B A A B A B A A B ... which bears some resemblance to the original string, but it's not just simply replacing 1/0 with A/B. On 6/23/12, Mike Stay <metaweta@gmail.com> wrote:
On Fri, Jun 22, 2012 at 9:40 PM, Robert Munafo <mrob27@gmail.com> wrote:
By step 49720, it has written: 1101011011010110101101101011011010110101101101011010110110101101101011010110110101101101011010110110101101011011010110110101101011011010110101101101011011010110101101101011011010110101101101011010110110101101101011010110110101101101011010110110101101011011010110110101101011011010110101101101011011010110101101101011011010110101101101011010110110101101101011010110110101101011011010110110101101011011010110110101101011011010110101101101011011010110...
It looks remarkably like this sequence that grows like Fibonacci's rabbits: 0 1 10 101 10110 10110101 1011010110110 101101011011010110101 ...
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