Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients

ABSTRACT

Taylor's law (TL) predicts that the variance and the mean will be related empirically through a power-law function. TL previously has been shown to arise even in the absence of biological, ecological or physical processes. We report here that the mean and variance of 110 finite integer sequences in the On-Line Encyclopedia of Integer Sequences (OEIS) obey TL approximately. We also show that the binomial coefficients on each row of Pascal's triangle obey TL asymptotically. These applications of TL to seemingly unrelated mathematical structures tend to confirm there might be purely statistical, context-independent mechanisms at play. Supplementary materials for this article are available online.

KEYWORDS:

• Law of the mean
• Power law
• Pascal's triangle
• Variance

Acknowledgment

The author wishes to thank the Editor and Associate Editor for their helpful comments. The author also gratefully acknowledges the contribution of the anonymous referee who pointed out a key mistake in an earlier version of the manuscript.

Notes

1 Deshouillers, Hennecart, and Landreau (2000) conjectured that A022566 contains a total of 113,936,676 terms with the largest being 7,373,170,279,850.