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**
Irrationality of Growth Constants Associated with Polynomial Recursions
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Stephan Wagner

Department of Mathematics

Uppsala University

Box 480

751 06 Uppsala

Sweden

and

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

Austria

**Abstract:**

We consider integer sequences that satisfy a recursion of the form
*x*_{n+1} = *P*(*x*_{n})
for some polynomial *P* of degree *d* > 1. If such a
sequence tends to infinity, then it satisfies an asymptotic formula of the
form *x*_{n} ∼ *A* α^{dn}, 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
(*G*_{n})_{n ≥ 0} satisfies an
asymptotic formula of the form *G*_{n} = *A*
α^{n} + *B* + O(α^{-ε
n}), 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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