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Variance Functions of Asymptotically Exponentially Increasing Integer Sequences Go Beyond Taylor's Law
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Joel E. Cohen

The Rockefeller University

1230 York Avenue, Box 20

New York, NY 10065

USA

**Abstract:**

Fibonacci, Lucas, Catalan, and all asymptotically exponentially increasing
positive sequences have counting functions (number of elements that do
not exceed a large number *y*) that are asymptotically proportional to
the logarithm of *y*, a slowly varying function. For all such sequences,
the variance of the elements that do not exceed *y* is asymptotically
proportional to the product of three factors: the logarithm of the
largest sequence element *a*(*n*) that does not exceed *y*; an explicit
function of the asymptotic ratio of successive sequence elements; and
the square of the mean of the elements that do not exceed *y*. The variance
function of an integer sequence has number-theoretic interest because it
distinguishes integer sequences according to the form of their variance
function. The variance function is also important in the analysis of
variance. Number-theoretic examples make it possible to analyze the
variance function of well specified processes observed without error.

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(Concerned with sequences
A000032
A000045
A000108
A014137
A094639.)

Received July 7 2022; revised versions received July 11 2022; July 19 2022; October 28 2022; November 7 2022.
Published in *Journal of Integer Sequences*,
November 8 2022.

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