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On Some Conjectures Concerning Stern's Sequence and Its Twist
Published/Copyright:
August 4, 2011
Abstract
The Stern sequence (also known as Stern's diatomic sequence) 
is defined by s(0) = 0, s(1) = 1, and for all n ⩾ 1 by
s(2n) = s(n), s(2n + 1) = s(n) + s(n + 1).
In a recent paper, Roland Bacher introduced the twisted Stern sequence
given by the recurrences t(0) = 0, t(1) = 1, and for n ⩾ 1 by
t(2n) = –t(n), t(2n + 1) = –t(n) – t(n + 1).
Bacher conjectured three identities concerning Stern's sequence and its twist. In this paper, we prove Bacher's conjectures.
Keywords.: Stern Sequence; Mahler-Type Functional Equations
Received: 2010-10-20
Revised: 2011-04-09
Published Online: 2011-08-04
Published in Print: 2011-December
© de Gruyter 2011
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Articles in the same Issue
- Completely Multiplicative Automatic Functions
- The k-Periodic Fibonacci Sequence and an Extended Binet's Formula
- Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis
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- On Some Conjectures Concerning Stern's Sequence and Its Twist
- Number of Weighted Subsequence Sums with Weights in {1, –1}
- Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences
- Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity
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- A Remark on a Paper of Luca and Walsh
- On the Tennis Ball Problem
- On the Conditioned Binomial Coefficients
- Convolution and Reciprocity Formulas for Bernoulli Polynomials
- Counting Finite Languages by Total Word Length
