OEIS Contributions

Updated October 21, 2026. This portfolio covers 17 published sequences: 11 authored and six extended. The A397075 formula update is now approved. One new sequence and two updates to existing entries remain in editorial review; the October contributions are listed separately below.

Why these. These sequences have nothing in common at the level of subject. They range over Ramsey theory, extremal graph theory, knot theory, additive combinatorics, Boolean function complexity, combinatorial game theory, and the algebra of graphs. What they share is how they were found. The work combines SAT solving, exact enumeration, and proofs: sometimes extending an existing sequence, sometimes organizing a count into a new entry, and sometimes finding a formula. A new OEIS entry is not necessarily a new mathematical result; references and credit belong with the individual contribution. The spread across fields is deliberate. And the method has a direction. The computed values are where I go looking for structure. The Rado numbers began as a column of computed integers and ended as a proved theorem; the rest are trailheads, some already climbed, others marked for whoever arrives next. The OEIS is where the results live, as permanent reference points anyone can build on or check.

Author search: oeis.org/search?q=alex+towell.

Authored sequences

These published entries list me as their author. Later contributions and editorial improvements are credited in the OEIS histories.

A394445 is backed by a closed-form theorem (a proof that the distinct 2-color Rado number for x+y=kz follows an explicit parity-dependent formula for all k >= 8), with a 500-term b-file. A394661 and A395644 come from a census of the KnotInfo prime-knot tables; A395521 from an exhaustive sandpile-group computation. A396815 and A396816 count the largest subset of Z/nZ partitionable into two (resp. three) sum-free sets, computed by SAT and checked with a second solver; Andrei Zabolotskii’s observation on A396815 turned into a small theorem, that its odd-indexed values are exactly twice the maximal sum-free set size (A211316). The later extension a(41)=28 was approved July 3, 2026, and was also checked using that identity.

A397074, A397075, and A397076 were originally published through five variables, where there are 616,126 NPN-equivalence classes. A397075 now extends farther, with the additional terms credited to Vladeta Jovovic. Each sequence counts the classes whose sensitivity, real polynomial degree, or decision-tree complexity is exactly the number of variables – three different ways a Boolean function can be “as complex as possible” in its arguments. Fresh orbit enumeration checks the counts through four variables. At five variables, recomputing the measures on the stored representative inventory checks the counts, but does not independently re-establish that inventory’s orbit completeness. The reference class totals are A000370.

A397641 and A397657 were published July 19, 2026. For A397641, complexes have positive total stoichiometry at most m, and reactions are directed pairs of distinct complexes. Networks must use both species; equivalence is only under the global species swap. Burnside counting and a separate two-species coefficient formula reproduce all 34 published terms. The complete m=2 row also agrees with direct enumeration of all reaction subsets.

A397657 counts labeled functions, with tail length measured as distance to the eventual cycle. Its second statistic counts cyclic points, not connected components. The generating-function calculation agrees cell-by-cell with enumeration of every function through n=6, covering all 91 published terms. Its marginals recover A216242 and A066324.

Extended sequences

Classical Zarankiewicz-problem sequences (originally by N. J. A. Sloane) where I computed and added new terms past the known frontier.

These thresholds are one greater than the corresponding extremal Zarankiewicz numbers z(m,n;3,4). Only the oriented 3x4 block is forbidden; a 4x3 block is allowed. The table records my contributions, not subsequent extensions by others.

The Zarankiewicz extensions use SAT: a satisfying assignment exhibits a dense matrix (a lower bound), and refuting the next density gives the matching upper bound, provided the encoding and symmetry reduction are sound. Agreement between solvers is distinct from independently checking a proof certificate. My later terms used a double-lex symmetry-breaking encoding to make the upper-bound proof tractable: the plain encoding could not settle z(10,11;3,4) in seven days, while the symmetry-broken version proved it in 27 minutes.

Extended: combinatorial game theory and Ramsey (July 2026)

Three further extensions, submitted June 30, 2026, and published July 2, 2026. They carry the method into combinatorial game theory, Sprague-Grundy values and Sylver coinage, alongside one more Ramsey number.

The game computations were cross-checked against the preceding published terms. A006672 a(11)=16 is pinned on both sides: a 4-regular C_4-free graph on 15 vertices gives the lower bound, and a Reiman / Kovari-Sos-Turan pair-counting argument forbids the matching upper bound, so the value is exact. A046671 is the subtlest – the Sylver-coinage Grundy value G(3,n) is a finite numerical-semigroup game whenever n is coprime to 3, which is exactly what makes a(17) computable even though Sylver coinage is undecidable in general.

A006672 has since been extended through n=38 by Leo Y. Zhang; my contribution listed here remains a(11)=16. The local continuation of A046671 beyond n=17 has not been submitted and is not included in this page’s published results.

October 2026 contributions — status checked October 21

The A397075 update has been approved; the other three contributions remain proposed. The revision links distinguish those editorial outcomes. Proposed means submitted and awaiting an editor, not accepted.

For A397075, XOR with input parity sends full real-degree functions to unbalanced functions and respects NPN equivalence. An independent Burnside calculation matches the source counts through n=10. Vladeta Jovovic had already proposed an overlapping extension and a b-file labeled through n=11, now published, so I did not submit my overlapping terms or claim credit for that extension. I have not independently checked his n=11 term.

For A397074, the statement is that s(f)=n, bs(f)=n, and C(f)=n are equivalent for every Boolean function on n variables, n >= 1. It does not say the three measures are equal in general.

A401195 counts graph isomorphism classes, unlike A395521, which counts group isomorphism types. The trivial group is included as cyclic. The original computation checked all 12,112 connected unlabeled simple graphs on two through eight vertices using two Smith-normal-form implementations, different deleted roots, and determinant checks.

The nine-vertex terms for both sandpile sequences come from one census of all 261,080 connected graphs on nine vertices. The graph list generated with nauty’s geng was matched, canonical form by canonical form, against Brendan McKay’s published inventory before any arithmetic ran. Each graph’s sandpile group was then computed with two independent Smith-normal-form implementations at different deleted roots, with the group order checked against a determinant. The census is archived on Zenodo as Two exact statistics of graph sandpile groups at nine vertices. Alper Ferudun has since posted an independent check of a(2)-a(9) for A395521 in its OEIS discussion. Earlier exhaustive Jacobian censuses, including Clancy, Leake, and Payne (2013), already reach nine vertices; this is not a claim to the first such census.

Publication notes

The three Boolean-measure entries were submitted June 15 and became public June 24, 2026. A396815 and A396816 were initially approved June 15. All seven Zarankiewicz terms listed above are published, as is the A394445 500-term b-file. The October updates, including the two proposed nine-vertex terms, do not change the total of 17 published sequences.

Reviewers

The published contributions benefited from review by OEIS editors including Michel Marcus, Sean A. Irvine, Max Alekseyev, and Jon E. Schoenfield.

Reproduce

The portfolio repository collects the contribution index, data, and reproduction scripts. Reproduction scope varies: some checks enumerate objects afresh, some recompute invariants on a stored census, and some verify a formula or a witness. Solver results and independently checked proof certificates are different kinds of evidence. Dependencies, input data, and expensive computation limits matter.

The local bundles record those distinctions and submission history. Recent local verification and submission-tracking updates have not yet been pushed to the public repository; the linked OEIS revision records above identify what was actually proposed.

The OEIS has been a shared ledger of integer sequences since 1964. Adding a term is a small, permanent contribution to a commons that outlasts any one project: a number that was unknown is now known, checked, and citable by anyone who needs it.