Vol. 56, No. 2, 1975

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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
Abstract

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

 (r)       (r)    n∏  (r)
b0  = 1,  bn+1 =   bi  + r for n ≧ 0.
i=1
(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

 (r)    2n
bn+1 ∼ 𝜃  , n → ∞,
(2)

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

b(n2+)1 = 22n + 1, n ≧ 0,
(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

       n     n
b(4n)= τ2 + τ− 2 + 2, n ≧ 1,
(4)

where τ = (√-
5 + 1)2 = 1.618 is the “Golden Mean”, and thus 𝜃(4) = τ2 = (√ -
5 + 3)2 = 2.618 .

Mathematical Subject Classification
Primary: 10A35
Milestones
Received: 7 March 1974
Published: 1 February 1975
Authors
Joel N. Franklin
Solomon Wolf Golomb