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I'm working on a project that uses strings of integers with the property that the numbers 1 though N are each used twice such that each pair of numbers X are X spaces apart.

For example, in the string:

3 1 1 3 5 7 4 8 6 5 4 2 7 2 6 8

The 1's are 1 space apart, the 2's are 2 spaces apart, the 3's are 3 spaces apart, etc.

I believe I've found the number of unique such strings for the following values of N

N : # of strings

2 : 0
3 : 0
4 : 6
5 : 10
6 : 0
7 : 0
8 : 504
9 : 2656
10 : 0
11 : 0
12 : 455936

I was hoping someone could tell me if someone else has studied these patterns? And if so, could point me in the right direction?

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For such things the "The On-Line Encyclopedia of Integer Sequences" is really handy, it gives this result for your sequence: oeis.org/A004075 –  Marcel Bischoff Aug 11 '11 at 18:03

2 Answers 2

up vote 8 down vote accepted

What you are describing are known as Langford sequences. An Internet search will give you http://legacy.lclark.edu/~miller/langford.html and other links.

Skolem or near Skolem sequences may also be of interest to you. I have a specialization of this I am studying: see Has anyone seen this version of ring toss (combinatorial object) before? .

Gerhard "Yes, Number Theory Is Involved" Paseman, 2011.07.23

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One of the recent volumes of Knuth's "Art of Computer Programming" (maybe volume 4), has these sequences and some things like a generating function. As far as I know, the asymptotic behaviour is not known.

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Welcome back to MathOverflow! Gerhard "Apologies For Missing You Before" Paseman, 2011.08.11 –  Gerhard Paseman Aug 11 '11 at 7:31

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