Binary Sequence Puzzle
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From: Leroy Quet on 24 May 2010 14:51 I actually hate those "guess the rule of the sequence of numbers" puzzles, especially since there are an infinite number of rules that work no matter the finite sample of the sequence given. But I bet that this would be a good puzzle anyway. What rule best determines the pattern of 0's and 1's in this infinite sequence? (Yes, all terms are either 0 or 1.) I think some people might get this. The beginning of the sequence is: 1,1,0,1,1,0,0,1,1,0,1,1,1,0,1,1,1,0,0,0,1,1,0,0,1, 1,0,1,0,1,1,0,1,1,1,0,0,1,1,0,1,1,1,1,0,1,1,1,1,1, 0,0,0,0,1,0,0,0,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0,1,1, 0,1,0,1,1,0,1,1,0 (I will submit this sequence to the On-LIne Encyclopedia Of Integer Sequences someday soon, because it is not in that database yet.) Thanks, Leroy Quet
From: Leroy Quet on 28 May 2010 08:43 Answer below: Leroy Quet wrote: \ I actually hate those "guess the rule of the sequence of numbers" \ puzzles, especially since there are an infinite number of rules that \ work no matter the finite sample of the sequence given. But I bet that \ this would be a good puzzle anyway. \ \ What rule best determines the pattern of 0's and 1's in this infinite \ sequence? (Yes, all terms are either 0 or 1.) I think some people \ might get this. \ \ The beginning of the sequence is: \ \ 1,1,0,1,1,0,0,1,1,0,1,1,1,0,1,1,1,0,0,0,1,1,0,0,1, \ 1,0,1,0,1,1,0,1,1,1,0,0,1,1,0,1,1,1,1,0,1,1,1,1,1, \ 0,0,0,0,1,0,0,0,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0,1,1, \ 0,1,0,1,1,0,1,1,0 \ \ \ (I will submit this sequence to the On-LIne Encyclopedia Of Integer \ Sequences someday soon, because it is not in that database yet.) \ \ Thanks, \ Leroy Quet a(1)=1. For n >= 2, if a(n) = 1, then append to the sequence the digits of binary n+1 (most significant digits first and least significant digits last). If a(n) = 0, then append to the sequence the digits of binary n+1 in reverse order (least significant digits first and most significant digits last). Here is the sequence broken up such that it is easier to see what I mean. Note that line-n is n written in binary either forwards or backwards. 1, 1,0, 1,1, 0,0,1, 1,0,1, 1,1,0, 1,1,1, 0,0,0,1, 1,0,0,1, 1,0,1,0, 1,1,0,1, 1,1,0,0, 1,1,0,1, 1,1,1,0, 1,1,1,1, 1,0,0,0,0, 1,0,0,0,1, 1,0,0,1,0, 1,1,0,0,1, 0,0,1,0,1, 1,0,1,0,1, 1,0,1,1,0 (I have submitted this to the Encyclopedia of Integer Sequences. But the sequence has yet to appear, but it should one of these days soon.) Thanks, Leroy Quet
From: Chip Eastham on 28 May 2010 09:55 On May 28, 8:43 am, Leroy Quet <qqq...(a)mindspring.com> wrote: > Answer below: > > Leroy Quet wrote: > > \ I actually hate those "guess the rule of the sequence of numbers" > \ puzzles, especially since there are an infinite number of rules that > \ work no matter the finite sample of the sequence given. But I bet > that > \ this would be a good puzzle anyway. > \ > \ What rule best determines the pattern of 0's and 1's in this > infinite > \ sequence? (Yes, all terms are either 0 or 1.) I think some people > \ might get this. > \ > \ The beginning of the sequence is: > \ > \ 1,1,0,1,1,0,0,1,1,0,1,1,1,0,1,1,1,0,0,0,1,1,0,0,1, > \ 1,0,1,0,1,1,0,1,1,1,0,0,1,1,0,1,1,1,1,0,1,1,1,1,1, > \ 0,0,0,0,1,0,0,0,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0,1,1, > \ 0,1,0,1,1,0,1,1,0 > \ > \ > \ (I will submit this sequence to the On-LIne Encyclopedia Of Integer > \ Sequences someday soon, because it is not in that database yet.) > \ > \ Thanks, > \ Leroy Quet > > a(1)=1. For n >= 2, if a(n) = 1, then append to the sequence the > digits of binary n+1 (most significant digits first and least > significant digits last). If a(n) = 0, then append to the sequence the > digits of binary n+1 in reverse order (least significant digits first > and most significant digits last). > > Here is the sequence broken up such that it is easier to see what I > mean. Note that line-n is n written in binary either forwards or > backwards. > 1, > 1,0, > 1,1, > 0,0,1, > 1,0,1, > 1,1,0, > 1,1,1, > 0,0,0,1, > 1,0,0,1, > 1,0,1,0, > 1,1,0,1, > 1,1,0,0, > 1,1,0,1, > 1,1,1,0, > 1,1,1,1, > 1,0,0,0,0, > 1,0,0,0,1, > 1,0,0,1,0, > 1,1,0,0,1, > 0,0,1,0,1, > 1,0,1,0,1, > 1,0,1,1,0 > > (I have submitted this to the Encyclopedia of Integer Sequences. But > the sequence has yet to appear, but it should one of these days soon.) > > Thanks, > Leroy Quet Hi, Leroy: I think you meant to say "For n >= 1..." since the first "string" to append are bits of n+1 = 2 (after the "basis step" a(1) = 1). regards, chip
From: Leroy Quet on 28 May 2010 10:12 Chip Eastham wrote: > On May 28, 8:43 am, Leroy Quet <qqq...(a)mindspring.com> wrote: > > Answer below: > > > > Leroy Quet wrote: > > > > \ I actually hate those "guess the rule of the sequence of numbers" > > \ puzzles, especially since there are an infinite number of rules that > > \ work no matter the finite sample of the sequence given. But I bet > > that > > \ this would be a good puzzle anyway. > > \ > > \ What rule best determines the pattern of 0's and 1's in this > > infinite > > \ sequence? (Yes, all terms are either 0 or 1.) I think some people > > \ might get this. > > \ > > \ The beginning of the sequence is: > > \ > > \ 1,1,0,1,1,0,0,1,1,0,1,1,1,0,1,1,1,0,0,0,1,1,0,0,1, > > \ 1,0,1,0,1,1,0,1,1,1,0,0,1,1,0,1,1,1,1,0,1,1,1,1,1, > > \ 0,0,0,0,1,0,0,0,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0,1,1, > > \ 0,1,0,1,1,0,1,1,0 > > \ > > \ > > \ (I will submit this sequence to the On-LIne Encyclopedia Of Integer > > \ Sequences someday soon, because it is not in that database yet.) > > \ > > \ Thanks, > > \ Leroy Quet > > > > a(1)=1. For n >= 2, if a(n) = 1, then append to the sequence the > > digits of binary n+1 (most significant digits first and least > > significant digits last). If a(n) = 0, then append to the sequence the > > digits of binary n+1 in reverse order (least significant digits first > > and most significant digits last). > > > > Here is the sequence broken up such that it is easier to see what I > > mean. Note that line-n is n written in binary either forwards or > > backwards. > > 1, > > 1,0, > > 1,1, > > 0,0,1, > > 1,0,1, > > 1,1,0, > > 1,1,1, > > 0,0,0,1, > > 1,0,0,1, > > 1,0,1,0, > > 1,1,0,1, > > 1,1,0,0, > > 1,1,0,1, > > 1,1,1,0, > > 1,1,1,1, > > 1,0,0,0,0, > > 1,0,0,0,1, > > 1,0,0,1,0, > > 1,1,0,0,1, > > 0,0,1,0,1, > > 1,0,1,0,1, > > 1,0,1,1,0 > > > > (I have submitted this to the Encyclopedia of Integer Sequences. But > > the sequence has yet to appear, but it should one of these days soon.) > > > > Thanks, > > Leroy Quet > > Hi, Leroy: > > I think you meant to say "For n >= 1..." > since the first "string" to append are > bits of n+1 = 2 (after the "basis step" > a(1) = 1). > > regards, chip Arrgg. You are right. (Bashes head into wall a dozen times.) I'll have to submit a correction to the EIS after that sequence and a related sequence appear. Thanks, Leroy Quet
From: Ludovicus on 28 May 2010 13:35
On 28 mayo, 10:12, Leroy Quet <qqq...(a)mindspring.com> wrote: > \ > What rule best determines the pattern of 0's and 1's in this > infinite sequence. As a Fourier Series can determine any pattern, you must to specify what type of function is permissible. |