The numbers you see to the right form a northwest corner of the Wythoff array:
Interestingly, the Wythoff array (1) contains every positive integer exactly once, (2) has increasing rows and columns, and (3) has interspersed rows - i.e., once the first term of any row lies between two consecutive terms of any other row, the alternating between the two rows continues forever. The three properties define an |

Now, suppose X = ( x(1), x(2), x(3), . . . ) is an increasing sequence of positive integers, with x(1) > 1. Write 1,2,3,...,30 across the top of a piece of paper, and then beneath each of these n, write the number x(n). Then write out successive rows of an array as follows:

Row 1 consists of

1, x(1), x(x(1)), x(x(x(1))), . . .

Let i be the least positive integer not in row 1, and write row 2 as

i, x(i), x(x(i)), x(x(x(i))), . . .

Let j be the least positive integer not in row 1 or row 2, and write row 3 as

j, x(j), x(x(j)), x(x(x(j))), . . .

Continue indefinitely, obtaining a *dispersion*, namely that of the sequence x. In the article

**C. Kimberling**, "Interspersions and dispersions," *Proceedings of the American Mathematical Society* 117 (1993) 313-321,

it is proved that every interspersion is a dispersion, and conversely. Special cases are discussed in

**N. J. A. Sloane,** Classic Sequences in the *Online Encyclopedia of Integer Sequences*

**A. Fraenkel and C. Kimberling**, "Generalized Wythoff arrays, shuffles and interspersions," *Discrete Mathematics* 126 (1994) 137-149,

**C. Kimberling**, "Stolarsky interspersions," *Ars Combinatoria* 39 (1995) 129-138,

**C. Kimberling**, "The first column of an interspersion," *Fibonacci Quarterly* 32 (1994) 301-314.

**C. Kimberling and J. Brown**, "Partial complements and transposable dispersions", *Journal of Integer Sequences* 7 (2004) 147-159. Click
**here** for a pdf file.

Interspersions are closely related to *fractal sequences*.

Fractal Sequences

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