bibkey: kagey2020a334595
authors: Peter Kagey
year: 2020
title: "OEIS A334595: the right edge of a binary XOR triangle"
doi: null
url: https://oeis.org/A334595
claim: "Conjecture: a(n) = 1 if and only if n is a power of two."
strata_touched:
  - D5/S3/Combinatorics/XorTriangle/ValueOne
license: CC-BY-SA-4.0
triage: anchor

Kagey, OEIS A334595

Revision 18 names the sequence:

Binary interpretation of the right diagonal of the XOR-triangle with first row generated from the binary expansion of n.

Its first comment defines the recurrence:

An XOR-triangle is an inverted 0-1 triangle formed by choosing a top row and having each entry in the subsequent rows be the XOR of the two values above it.

The fourth %C comment, the entry’s second conjecture, reads:

Conjecture: a(n) = 1 if and only if n is a power of two.

The example fixes the edge order:

Reading the right side of the triangle starting from the upper-right corner gives 10100 which is the binary representation of 20 = a(19).

The module uses the unpadded most-significant-first binary expansion of each positive natural number. Each subsequent row uses ordinary adjacent Boolean XOR. All right-edge positions are retained, including leading zeros, from the top-right corner to the apex. The exact assertion is, for every n >= 1, a(n) = 1 if and only if there exists a natural k with n = 2^k. The exponent zero includes n = 1. The record-position conjecture is separate.

The quoted source text and its adapted encoding are attributed to Peter Kagey and the OEIS Foundation under CC BY-SA 4.0, as specified by the OEIS end-user license. The adaptation expresses the row, recurrence, orientation and binary interpretation in Lean and Scribe; it reproduces no source program or table.

Verified locator

• Source: https://oeis.org/A334595, revision 18.
• Assertion: fourth %C comment, second conjecture.
• Encoding: %N, first %C comment and the n = 19 example.