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Topic: Inverse Stolarsky array
Replies: 1   Last Post: Jan 28, 2003 8:55 AM

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Clark Kimberling

Posts: 20
Registered: 12/3/04
Inverse Stolarsky Array
Posted: Jan 28, 2003 8:55 AM
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Here's an answer to your question (a year later), starting with the
"practical way" to create the "dispersion" of a sequence, quoted from

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 x(i) be the least positive integer not in row 1, and write row 2

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

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

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

Continue indefinitely, obtaining a dispersion. The term "dispersion"
was introduced in the article cited in Online Encyclopedia of Integer
Sequences at A035507 (the inverse Stolarksy array).

(end of quote)

Now, given the Stolarsky array, let {x(i)} be the sequence consisting
of the first column after the first term is removed. That is,
x(i) starts with 4,7,9,12,14,17,20,...

When you apply the above method to this sequence, the resulting array
is the inverse Stolarsky array.

The main thing is this: if you start with an array A(i,j) that is a
dispersion (hence also an interspersion), then the inverse of the
array is also a dispersion. Also, of course, the inverse of the
inverse of A is A.

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