11
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What is the next number in this never-ending sequence?

          0 
          1 
          1 
        110 
        111 
        101 
      11000 
      11101 
      10101 
    1100110 
    1001011 
  110101001 
  110010000 
  100111001 
11010001001

What were the previous two numbers?

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  • $\begingroup$ Could it be the case that rot13 (Bar ebj vf zvffvat va gur frdhrapr va gur dhrfgvba)? $\endgroup$ – melfnt Jan 29 at 16:24
  • $\begingroup$ @melfnt Pretty sure not; see my answer. Incidentally, if you omit the space between "rot13" and the parentheses, then you will make things easier for users of the userscript that provides tooltips for rot13'ed text on Stack Exchange pages. $\endgroup$ – Gareth McCaughan Jan 29 at 16:28
  • $\begingroup$ Sorry it doesn't let me edit my comment I don't know why $\endgroup$ – melfnt Jan 29 at 16:48
  • $\begingroup$ @melfnt all rows are there. (Comments can only be edited for 5 minutes after posting.) $\endgroup$ – Bass Jan 29 at 18:14
5
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The next number is

11110100110

and the previous two were

11 and then 1.

These are

the Fibonacci numbers in base -2.

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  • $\begingroup$ Exactly so. I kind of figured you'd recognise both the sequence and the notation on sight; I was hoping someone might work out the arithmetics pattern before that, but alas, no such luck :-) For bonus points, if the next word in the title were "four," what would the five words after that be? $\endgroup$ – Bass Jan 29 at 17:40
  • $\begingroup$ I didn't actually recognize the notation on sight, though once it occurred to me I felt stupid for not thinking of it faster. The connection with the rot13(svobanppv ahzoref) was obvious from the outset, though. For your bonus question, the title could continue with "rot13(fgrcf sbejneq, naq fb ba)", though maybe you have something more ingenious in mind? $\endgroup$ – Gareth McCaughan Jan 29 at 18:48
  • $\begingroup$ No, nothing more complex than that, bonus points awarded; just felt that the title's triple meaning wordplay could use some more attention, what with me feeling pretty smug about it and all. :-) $\endgroup$ – Bass Jan 29 at 19:14

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