A function-theoretic approach to the study of nonlinear recurring sequences

Joel N. Franklin and Solomon Wolf Golomb

Vol. 56 (1975), No. 2, 455–468

DOI: 10.2140/pjm.1975.56.455

Abstract

For every real r ≧ 0, there is a sequence {b_n^(r)} defined by

(1)

These sequences were considered previously, in [1], for integer values of r, and it was shown that there is a constant 𝜃 = 𝜃(r) such that

(2)

for each r = 1,2,3,⋯ . It was observed that

(3)

whereby 𝜃(2) = 2, and the problem was proposed “to determine the algebraic or transcendental character of the real numbers 𝜃(r) for r = 1,3,4,5,6,⋯.”

In this paper, we observe explicitly (in §II) that

(4)

where τ = ( + 1)∕2 = 1.618⋯ is the “Golden Mean”, and thus 𝜃(4) = τ² = ( + 3)∕2 = 2.618⋯ .

Mathematical Subject Classification

Primary: 10A35

Milestones

Received: 7 March 1974

Published: 1 February 1975