Journal of Integer Sequences, Vol. 25 (2022), Article 22.9.3

Variance Functions of Asymptotically Exponentially Increasing Integer Sequences Go Beyond Taylor's Law

Joel E. Cohen
The Rockefeller University
1230 York Avenue, Box 20
New York, NY 10065


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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