The sequence A000975 in OEIS can be defined by A1 = 1, An+1 = 2An if n is odd, and An+1 = 2An + 1 if n is even. This sequence satisfies other recurrence relations, admits some closed formulas, and is known to enumerate several interesting families of objects. We provide a new interpretation of this sequence using a binary operation defined by a⊖ b := −a− b. We show that the number of distinct r...

A binary reflected Gray code (Gray code for short) is a one-to-one function G(i) of the integers 0 • i • 2 – 1 (for some pre-selected value of n) to binary numbers such that the binary number for G(i) and G(i+1) differ in exactly one bit. For example, with N = 3, the Gray codes (written in binary) are 000, 001, 011, 010, 110, 111, 101, 100. In this paper, we describe a dynamic programming algor...

We consider two quantities that measure complexity of binary strings: KM(x) is defined as the minus logarithm of continuous a priori probability on the binary tree, and K(x) denotes prefix complexity of a binary string x. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence ω such that the sum of 2KM(x)−K(x) over all prefixes x of ω i...