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Cyborg Mathematics

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Integer sequences (OEIS)

The OEIS is the reference library of integer sequences that mathematicians use to recognise a list of numbers. Every new entry and every new term is reviewed by a volunteer editor before it is published, so each item here has passed at least one outside check.

13 results · 3 articles

Results

Numbered by their place in the strongest-first list

  1. 02Preprint, not refereed

    My OEIS counts of dependency charts, and the conjecture from them that Roberto Blanco Gómez proved

    A dependency chart (a labeled DAG) is a set of tasks in which some tasks wait on others and nothing waits on itself, even indirectly. I counted the charts in which each task waits on at most two others, up to tasks, and at most three, up to . These were the first OEIS entries for these counts, and the only earlier table, from 2017, stopped at . I listed a pattern in their remainders as a conjecture in OEIS A397711, and in October 2026 Roberto Blanco Gómez proved it in a preprint posted on Zenodo, not yet refereed.

    Checked outside this site

    • PreprintRoberto Blanco Gómez’s proof, not yet refereed by a journal, . Zenodo preprint
    • OEIS editorSean A. Irvine approved his note of the proof on my entry, . OEIS A397711
    • OEIS editorSean A. Irvine approved the entry itself, . OEIS A397711
    SequencesGraphs and Erdős
  2. 03

    The take-away game 0.007, computed four times past its published table

    In a take-away game, two players take turns removing counters from heaps under a fixed rule, and a table of values says which moves win. In July 2026 my run computed that table for the game 0.007, known as Treblecross, for every heap size up to ▲, four times further than the table that had stood since 2021. Achim Flammenkamp, who keeps the reference tables, checked the program himself and reran the game over his old table’s range with his own programs, getting the same results, and he has since taken the game further with his own runs.

    Checked outside this site

    • RerunAchim Flammenkamp checked the program and reran the game over his old range, . his octal tables
    • OEIS editorPublished as a new OEIS entry, . OEIS A395248
    SequencesGames
  3. 04

    More take-away games extended, three of them rerun by the tables’ maintainer

    In July 2026 I extended the published tables of more of these games, ▲ in all counting 0.007, two to four times further than before. Flammenkamp reran three of them, 0.006, 0.606 and 0.114, over their full new ranges on his own machines with his own programs and got the same answers; with 0.007, he reran four of the games in all. In game 0.114 the runs also found more heap sizes with rare values. Rare values are the exceptions in a game’s table; if they ever stop appearing, the table is known to repeat forever, which is how such a game gets solved.

    Checked outside this site

    • RerunAchim Flammenkamp reran three of the games with his own programs. his octal tables
    • CreditHis tables added the extensions with credit by name, . the credit line
    • OEIS editorThe extended OEIS entries were published, the last of them on this date, . OEIS A071459
    SequencesGames
  4. 05

    A published upper bound on the domination number of the twelfth Fibonacci cube, improved and now stated in the OEIS

    A dominating set in a network is a set of points that every point is in or next to, and the domination number is the size of the smallest one. For the twelfth Fibonacci cube, a network of points, the domination number was known to lie between and (Azarija, Klavžar, Rho and Sim, 2018). I found a dominating set of size , checked by three separate programs, so it is at most . A month-long solver run that tried to settle the lower end stopped without an answer, and I published it as a documented attempt.

    Checked outside this site

    • OEIS editorMichael De Vlieger approved the comment and the attached dominating set, . OEIS A291295
    • ArchiveThe set, its checker and the solver run archived, . Zenodo DOI
    SequencesGraphs and Erdős
  5. 09

    The first OEIS counts of shapes built from Josehedron cells

    The Josehedron is a -faced solid whose copies fill space with no gaps. Its shape appeared, unnamed, in Moritz W. Schmitt’s 2016 survey of such solids, and Mathias Bernhard named it in 2026. I counted the shapes made by gluing copies of it (cells) face to face, up to cells, in the three standard ways: free (a shape, its turns and its mirror image count once), one-sided (mirror images count separately) and fixed (shapes count as the same only when one slides onto the other). The OEIS had none of these counts before; they are now A398957, A398958 and A398959.

    Checked outside this site

    • OEIS editorMichael De Vlieger approved A398958 and A398959, . OEIS A398958
    • OEIS editorA398957 published after review by further editors, . OEIS A398957
    ShapesSequences
  6. 10

    Shape counts for more space-filling cells, with an independent match

    From mid-October to early November 2026 the OEIS published sequences of mine counting the shapes built from space-filling cells, among them the cell of the Laves graph and one of Peter Engel’s -faced cells, the most faces known for a cell of this kind. For two of the cells an OEIS contributor, Alper Ferudun, computed further terms on his own, and they matched mine term for term.

    Checked outside this site

    • OEIS editorPublished by OEIS editors, the last four on this date, . OEIS A401463
    • MatchAlper Ferudun’s own terms matched mine, . OEIS A401256
    ShapesSequences
  7. 11

    Two OEIS sequences counting hexagon shapes without holes, extended

    Polyhexes are shapes made of hexagons joined edge to edge, the outlines that benzenoid molecules take. Both sequences count polyhexes without holes by perimeter, treating shapes that differ only by a turn as the same. A258204 counts them one-sided, so a shape and its mirror image count separately; A258205 counts only those with mirror symmetry. I extended each by ▲ terms, up to perimeter index ▲, where the one-sided count reaches ▲. Two separately written programs agree at every size, and an identity with published data pins down some of the values on its own.

    Checked outside this site

    • OEIS editorSean A. Irvine published A258204, . OEIS A258204
    • OEIS editorSean A. Irvine published A258205, . OEIS A258205
    ShapesSequences
  8. 12

    A fullerene table in the OEIS that an old bug fix never reached, corrected

    Fullerenes are cage molecules made of carbon, like the buckyball. The OEIS table that counts them by size took its larger values from a 2006 data file made before a bug in the counting program was fixed, so those values were too small. I replaced ▲ of them, of which had been wrong for about ▲ years.

    Checked outside this site

    • OEIS editorMichael De Vlieger published the fix and Andrei Zabolotskii reviewed it, . OEIS A057210
    SequencesChemistry
  11. 15

    New terms for a shape sequence marked hard to extend

    OEIS A385028 counts the shapes built by gluing copies of an eleven-faced space-filling solid, the bisymmetric hendecahedron, face to face, and it carries the OEIS mark for sequences that are hard to extend. I computed new terms and computed one of them again with a second, separately written program.

    Checked outside this site

    • OEIS editorPublished by an OEIS editor, . OEIS A385028
    ShapesSequences
  12. 16

    Shape counts for the sphenoid hendecahedron, a space-filling solid with no earlier counts I could find

    The sphenoid hendecahedron is an eleven-faced solid whose copies fill space. The only published description of how they pack was in words, so I turned it into exact positions and showed that it is the only packing in which every copy meets its neighbors in the same way. Then I counted the shapes built by gluing copies face to face, which had no earlier counts I could find. The argument is in the published entry.

    Checked outside this site

    • OEIS editorPublished as new OEIS entries, with their full data files, . OEIS A397708
    ShapesSequences

Articles

Notes on the numbers marked ▲

  • Heap sizes covered by my July 2026 run of the game 0.007 (about 1.07 billion): 2^30. This was four times the table that had stood since 2021, and the furthest published computation of the game when it finished in July 2026. Achim Flammenkamp's tables have since gone further: to 2^31, about 2.1 billion heap sizes, by September 2026 (his own run), and to 2^34, about 17 billion, as listed on his page (checked October 29, 2026). The furthest values are now on his page.
  • Unsolved take-away games whose published tables I extended in July 2026: 12. Achim Flammenkamp, who keeps the reference tables, has since taken every one of these games further, so his page (checked October 29, 2026) now has the furthest values.
  • New terms added to each of A258204 (one-sided holeless polyhexes) and A258205 (holeless polyhexes with bilateral symmetry), n = 20 through 32: 13. The new terms of both entries are published in the OEIS: A258204’s on 2026-09-08 and A258205’s on 2026-09-12.
  • Largest perimeter index n (perimeter 2n) now in A258204 (one-sided holeless polyhexes) and A258205 (holeless polyhexes with bilateral symmetry): 32. Both entries are now published in the OEIS through n = 32: A258204 on 2026-09-08 and A258205 on 2026-09-12, both approved by Sean A. Irvine.
  • One-sided holeless polyhexes of perimeter 64, counted up to rotation, mirror images counted separately (A258204 at n = 32): 750,157,868,462. A one-sided count: a shape and its mirror image count as two shapes unless they are the same.
  • Values of the OEIS fullerene table A057210, a(68) to a(100), replaced because its 2006 data file predated a program bug fix: 33. Of the values replaced, the ones at n = 69, 70, 71, 74 and 80 were already right.
  • Years, roughly, that the wrong values in A057210 stood (the data file dates from 2006): 20. An approximate span, from the 2006 data file to the August 2026 correction.
  • Largest n in the OEIS table of largest Sidon sets (A143824) after the extension: 586. The values are exact because optimal Golomb rulers are known through 28 marks; the table can go no further until a longer optimal ruler is found.