The Binary Two-Up Sequence

TL;DR

The paper introduces the Binary Two-Up Sequence, a unique sequence of integers with specific binary properties, providing an explicit formula and confirming that each nonzero term is at most the sum of two powers of 2.

Contribution

It decomposes the sequence into atomic patterns and derives an explicit formula, confirming a conjecture about the sum of powers of 2.

Findings

Sequence can be decomposed into atoms with specific binary patterns

Explicit formula for sequence terms derived

Every nonzero term is the sum of at most two powers of 2

Abstract

The Binary Two-Up Sequence is the lexicographically earliest sequence of distinct nonnegative integers with the property that the binary expansion of the n-th term has no 1-bits in common with any of the previous floor(n/2) terms. We show that the sequence can be decomposed into ``atoms'', which are sequences of 4, 6, or 8 numbers whose binary expansions match certain patterns, and that the sequence is the limiting form of a certain ``word'' involving the atoms. This leads to a fairly explicit formula for the terms, and in particular establishes the conjecture that every nonzero term is the sum of at most two powers of 2.