Experiments on the integer sequence A359012

A359012 consists of a sequence of numbers k that are a substring of xPy where k=concatenation(x,y) and xPy is the number of permutations.

If x,y,k are natural numbers and k is formed by the concatenation of the digits of x followed by those of y, i.e., k = concatenation(x,y), then k belongs to A359012, if xPy contains the number k.

The first eight numbers are given below:

• A359012.py: brute-force generator for the sequence plus helpers for derived annotations.
• main.py: exports the canonical sequence data to a single CSV file.
• A359012.csv: the source dataset of generated terms, digit-length annotations, and row-level analysis fields below 10^6.
• analyze_sequence.py: computes a compact mathematical summary from A359012.csv and writes ANALYSIS.md.
• index.html: an interactive D3-powered website for exploring the sequence visually.
• assets/site.js and assets/site.css: the browser-side logic and styling for the website.
• assets/logo.svg and assets/favicon.svg: the visual identity for the site and browser tab.
• tests/test_a359012.py: regression tests for known terms and basic generator behavior.

Using the current brute-force search bound 10 <= k < 10^6, the repository finds 712 terms.

• The sequence is still sparse in this larger range: only 712 terms appear below one million.
• Term lengths are now distributed as 4 three-digit terms, 29 four-digit terms, 114 five-digit terms, and 565 six-digit terms.
• Balanced longer splits dominate the current data, especially (3, 3) with 391 cases and (4, 2) with 162 cases.
• Substring hits are overwhelmingly internal: 3 prefix hits, 1 suffix hit, and 708 strictly internal hits.
• 708 of the 712 witness values xPy end in 0.
• Repeated-digit terms now dominate strongly: 572 terms have repeated digits, while 140 have all digits distinct.
• Palindromic terms do occur: 9999, 29092, 343343, and 805508.

See ANALYSIS.md for the generated summary.

Generate the CSV source:

python main.py

Generate the analysis summary from the CSV source:

python analyze_sequence.py

Preview the website locally:

python -m http.server

Then open http://localhost:8000.

Deploy the website with GitHub Pages:

• The workflow deploy-site.yml publishes the static site on pushes to main.
• The deployment uploads index.html, assets/, A359012.csv, and supporting markdown files as the Pages artifact.

Run the tests:

pytest -q

• How does the counting function grow with the search bound?
• Does the dominance of balanced splits persist for larger ranges?
• Can the trailing-zero bias be explained directly from the arithmetic of xPy?
• Are there infinite families of terms with repeated digits or special endings?

• John Samuel

• Zenodo
• Release Notes

All code are released under GPLv3+ licence. The associated documentation and other content are released under CC-BY-SA.