10
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0, 1, 11.2, 10.2, 113.2, 103, 103.2, 0.2, 1.2, 2.2, 2, ?

The only hint I'll give at this stage is that the sequence is infinite.

If nobody gets it within a few days, I'll give a few terms that come after the wanted term.

Have fun! :-)

Edit: this is a wholly mathematical puzzle, a number sequence. There are no red herrings, elements meant to mislead or be ignored, or punctuation tricks etc.

Note: this sequence can be defined in 12 words.

Further hint (in response to comments from @MohitJain and @MattMalone):

the numbers are all written in the same base, which isn't base 4 or 10.

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    $\begingroup$ It's very clear. Usefulness can only be judged if someone knows the answer or has got somewhere with it. See my comment on @Matt Malone's answer below $\endgroup$ – h34 Dec 28 '14 at 10:54
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    $\begingroup$ @warspyking - What do you mean by "way too broad"? The putting together of mathematical ideas to get this sequence is pretty cool, and the application of effort to solve it is something I would have thought many puzzlers of a mathematical bent would find useful. Might you delay forming an opinion on whether it should be allowed to stay up until people have had some more time to tackle it? $\endgroup$ – h34 Dec 29 '14 at 15:28
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    $\begingroup$ @warspyking - No offence but I think that point doesn't stand up the way you want. The next item of any sequence 'can' be anything, sure. Assuming you can't solve this puzzle, if you come up with a generating rule to prove your point (try it, with 57 as the missing term) I can guarantee it will be obviously arbitrary and contrived. Your position implies there is nothing useful, to an extent supporting a puzzle staying open here, in asking people to find the neat generating rule for a sequence and use it to calculate the next term. Because the next term can always be anything you want. Silly! $\endgroup$ – h34 Dec 29 '14 at 15:50
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    $\begingroup$ I generally try not to downvote unanswered questions because the answer matters (in puzzling challenges at least). My last question was pretty heavily downvoted. Then when the answer was posted the score climbed into positive territory and sits at 17 today. The answer revealed it to be a good question. Just because I haven't been able to find the answer doesn't make it a bad question. $\endgroup$ – Matt Malone Dec 29 '14 at 18:11
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    $\begingroup$ If you protest that there are infinitely many possible solutions, then find the one with lowest Kolgoromov complexity :) This puzzle might be brilliant and it might be terrible, but you can't write off all "find the sequence" puzzles like that. $\endgroup$ – Lopsy Dec 29 '14 at 18:13
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Based on the following analysis, the next number in the sequence is:

12.2

The number base is:

2i, also known as quater-imaginary

This table converts the numbers to base 10:

Table1

This graph shows the sequence:

as a spiral
Graph

Additional comments by h34

Here is the sequence defined in 12 words:

Gaussian integers, base 2i, ordered in counterclockwise Ulam spiral beginning at 0.

Anyone who hasn't encountered

Gaussian integers, base 2i or Ulam spirals

before will find a lot of interesting stuff by clicking on those links.

Or if you want a quick explanation here, here goes...

Gaussian integers are numbers of the form a+bi, where a and b are integers and i is the square root of minus 1. These were first studied by Carl Friedrich Gauss in 1832.

Base 2i means what it says: instead of expressing a number using digits intended to be multiplied by powers of 10 (giving the decimal system), do it for powers of 2i. Base 2i is also known as quater-imaginary. It was first described by Donald Knuth in 1955.

For expressing Gaussian integers, base 2i is symbolically less complex than any real base.

To express negative real numbers, it needs no symbols other than digits. Unlike a real base, it needs no minus symbol. To express Gaussian integers, it needs no symbols other than digits and, if necessary, a point to separate the units place from the $(2i)^{-1}$ place. And when a $(2i)^{-1}$ digit is needed, it's always 2 and it's the only digit after the point.

The decimal system needs no point for integers whether real or Gaussian, but for negative reals it uses a minus symbol; for Gaussian integers with imaginary part nonzero it needs i; and for Gaussian integers with both real and imaginary parts nonzero it needs a plus symbol, a minus symbol, or both.

Examples of numbers written in base 2i:

1+i is written 11.2 because 1+i = 1(2i) + 1(1) +2$(2i)^{-1}$ = 2i + 1 + 2(-i/2);

-1 is written 103 because -1 = 1$(2i)^2$ + 3(1) = -4 + 3

Expressed in base 2i, the non-negative integers run

0,1,2,3,10300,10301,10302,10303,10200,10201,10202,...,

and the negative integers, ordered by increasing magnitude, run

103, 102, 101, 100, 203, 202, 201, 200, 303, 302

Some numbers have more than one representation, e.g. 1/5 can be written as either 1.(0300) or 0.(0003). And a number such as 8 can be written 10200 or simply as 8. To get a unique representation of a Gaussian integer, just stipulate that all digits used must be between 0 and 3 inclusive.

Real numbers have their nonzero digits all in odd-order positions; pure imaginary numbers, all in even-order positions. So the expression for a+bi can be got from adding the expressions for a and b, and no 'carrying' is necessary.

The Ulam spiral was invented by Stanislaw Ulam in 1963. It's a way of writing down the positive integers on a square grid. Using standard Cartesian notation, write 1 at the origin (0,0), 2 at (1,0), 3 at (1,1), 4 at (0,1), 5 at (-1,1) etc. If you then mark the primes, you get a nice pattern, including for example a (for a time) almost-solid line of primes of the form $4x^2 + 2x + 41$ (related to Euler's polynomial $x^2 + x + 41$).

I used an Ulam spiral to order the Gaussian integers: 0, 1, 1+i, i, -1+i, -1 , etc. Then I expressed those numbers in base 2i to get the puzzle sequence. The missing number is the base 2i expression of 2+i, which is 12.2.

The next hints would have been:

The numbers that follow the missing term are 12, 11, 10, 113, 112, 112.2, 102, 102.2, 1132, 1133, 1030, ...

Each of the three main mathematical ideas used to construct this sequence is associated with a distinct mathematician with a surname containing 4 or 5 letters.

Whoever thunk of this won't maul or gas us. (Anagrams of the three mathematicians' surnames.)

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    $\begingroup$ Nailed it. That explains why all the S(n)-S(n+1) diffs were even numbered terms of the sequence itself: those were 1, -1, i, and -i. $\endgroup$ – Matt Malone Jan 1 '15 at 22:42
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    $\begingroup$ Nice. Always something new to learn on this site.... $\endgroup$ – BmyGuest Jan 1 '15 at 22:43
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    $\begingroup$ @Len - Brilliant - well done! In 12 words, the sequence lists Gaussian integers, base 2i, ordered in counterclockwise Ulam spiral starting at 0. I have got some more to say about this sequence, too long for a comment. Shall I post it in another answer? $\endgroup$ – h34 Jan 1 '15 at 23:29
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This answer is a work in progress. Below are the differences between terms n and n+1:

1, 10.2, -1, 103, -10.2, 0.2, -103, 1, 1, -0.2

These values are all even index terms from the original series S or their opposites. Like so:

S(2), S(4), -S(2), S(6), -S(4), S(8), -S(6), S(2), S(2), -S(8)

Still trying to find some pattern in these. Another thing I notice:

S(3) = S(2) + S(4)
S(5) = S(4) + S(6)
S(7) = S(6) + S(8)

Which is to say S(n) = S(n-1) + S(n+1) for n = 3, 5 and 7 but this pattern breaks down at 9. Somewhat interestingly S(9) = (S(8) + S(10)) / 2. Hmmm.

Update: Still trying. To state the above a little differently:

S(2) = S(3) - S(4)
S(4) = S(5) - S(6)
S(6) = S(7) - S(8)
S(8) = S(10) - S(11) ?! What happened to S(9)? Or...
S(8) = 2 * S(9) - S(10) as noted above

The three problems with this line of investigation:
1) The pattern disappears at the end
2) It can't predict odd-numbered terms
3) It's hard to ignore

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  • $\begingroup$ Keep going! The sequence has a neat one-sentence definition which brings together three mathematical ideas that as far as I'm aware haven't been brought together before (which is why I felt the downvote was a bit insulting). I think the kind of analysis you're doing is bound to take you forward. For that reason, the next hint will give a large number of terms, not just a few, which may help with the pattern of breakdown of the finer-grain pattern. $\endgroup$ – h34 Dec 28 '14 at 10:26
  • $\begingroup$ I'll start doing the situation riddle thing and ask, do the terms following the wanted number in the sequence ever involve "1" after the decimal point? $\endgroup$ – Joe Z. Dec 29 '14 at 21:44
  • $\begingroup$ No numbers in the sequence have a "1" after the point. $\endgroup$ – h34 Dec 30 '14 at 10:33

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