[Paper Review] An Exploration of Sequence A000975

Paul K. Stockmeyer|arXiv (Cornell University)|Aug 29, 2016

Algorithms and Data Compression1 references3 citations

TL;DR

This paper proves that all conjectured characterizations and open questions regarding OEIS sequence A000975—previously unverified—are mathematically correct. It establishes multiple equivalent formulations of the sequence, including recursive, binary, and closed-form expressions, and confirms its occurrence in diverse contexts such as the Chinese rings puzzle, Gray code distances, circular affinity group partitions, and 2-colored operads of colored bubbles, unifying these under a single combinatorial framework.

ABSTRACT

Sequence A000975 in the Online Encyclopedia of Integer Sequences (OEIS) starts out 1, 2, 5, 10, 21, 42, 85, ... . As of July 1, 2016, the description in the OEIS lists several characterizations of this sequence and numerous examples of instances where this sequence occurs. It also presents a "not yet proved" result, a conjecture, and an unanswered question concerning this sequence. In this paper we show that all of these proposed results are in fact true.

Motivation & Objective

• To verify the validity of all unproven conjectures and open questions regarding sequence A000975 in the OEIS as of July 2016.
• To unify and formally prove the equivalence of multiple existing characterizations of A000975, including recursive, binary, and closed-form definitions.
• To confirm the sequence's occurrence in diverse combinatorial contexts, such as the Chinese rings puzzle, Gray code distances, and circular affinity group partitions.
• To resolve a longstanding ambiguity in a 2-colored operad of bubbles by correcting flawed generating functions and deriving the correct count of based and unbased bubbles.
• To establish a one-to-one correspondence between affinity group partitions and colored bubbles in the operad framework.

Proposed method

• Derives multiple equivalent characterizations of A(n), including recursive definition, binary representation with alternating bits, and closed-form expressions involving floor functions and powers of two.
• Uses the identity A(n) = ⌊(2/3)·2ⁿ⌋ to unify the sequence with the binary expansion of 2/3, providing a novel derivation of the closed form.
• Applies modular arithmetic and combinatorial summation to prove S(n) + S(n+1) = 2ⁿ, where S(n) counts binary strings of length n with b-count ≡ (2n+1) mod 3.
• Constructs a bijection between affinity group partitions of n+2 people and based bubbles of arity n in a 2-colored operad, using edge coloring rules based on group transitions.
• Corrects erroneous generating functions in prior work by identifying swapped variables in numerators and deriving the correct 2-variable generating functions for based and unbased bubbles.
• Uses recurrence relations and structural constraints (e.g., existence of two consecutive same-colored edges) to define and count valid bubbles in the operad.

Experimental results

Research questions

• RQ1 Are all conjectured characterizations of sequence A000975 in the OEIS, including its recursive, binary, and closed-form definitions, mathematically equivalent and correct?
• RQ2 Does the sequence A(n) correctly count the number of moves in the n-ring Chinese rings puzzle?
• RQ3 Is the distance between all-0 and all-1 strings in the n-bit Gray code equal to A(n)?
• RQ4 Is the number of valid 3-partitions of n+2 people around a circular table into non-adjacent affinity groups equal to A(n)?
• RQ5 Do the counts of based and unbased bubbles of arity n in the 2-colored operad of bubbles match A(n−1), and are the generating functions correctly formulated?

Key findings

• All conjectured characterizations of sequence A000975 are proven equivalent, including the recursive definition, binary representation with alternating bits, and the closed form A(n) = ⌊(2/3)·2ⁿ⌋.
• The sequence A(n) correctly counts the number of moves required to solve the n-ring Chinese rings puzzle, confirming its role in this classic mechanical puzzle.
• The distance between the all-0 and all-1 strings in the n-bit binary reflected Gray code is exactly A(n), validating its occurrence in Gray code theory.
• The number of ways to partition n+2 people around a circular table into three non-adjacent affinity groups is A(n), confirming a result from Knuth and Lossers.
• The corrected generating functions for based and unbased bubbles in the 2-colored operad yield counts matching A(n−1), and a one-to-one correspondence is established between affinity group partitions and such bubbles.
• The number of valid based bubbles of arity n is A(n−1), and this count satisfies the recurrence A(n) = (2ⁿ−1) − A(n−1), confirming its consistency across combinatorial models.