Journal of Integer Sequences, Vol. 24 (2021), Article 21.1.6

Irrationality of Growth Constants Associated with Polynomial Recursions

Stephan Wagner
Department of Mathematics
Uppsala University
Box 480
751 06 Uppsala
Department of Mathematical Sciences
Stellenbosch University
Private Bag X1
Matieland 7602
South Africa

Volker Ziegler
Department of Mathematics
University of Salzburg
Hellbrunnerstrasse 34/I
5020 Salzburg


We consider integer sequences that satisfy a recursion of the form xn+1 = P(xn) for some polynomial P of degree d > 1. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form xnA αdn, but little can be said about the constant α. In this paper, we show that α is always irrational or an integer. In fact, we prove a stronger statement: if a sequence (Gn)n ≥ 0 satisfies an asymptotic formula of the form Gn = A αn + B + O(αn), where A, B are algebraic and α > 1, and the sequence contains infinitely many integers, then α is irrational or an integer.

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(Concerned with sequences A000058 A003095 A076949.)

Received August 12 2020; revised versions received August 13 2020; December 31 2020; January 1 2021. Published in Journal of Integer Sequences, January 2 2021.

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