OEIS Contributions
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. Each is a quantity with no known formula, computed exactly by SAT solving or exhaustive search, pushed one value past the published frontier, and checked against all prior art before it was submitted. The spread across fields is deliberate: it is evidence that the method travels. 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 sequences were created and first populated by me.
| Sequence | Description | Domain |
|---|---|---|
| A394445 | Distinct-variable 2-color Rado numbers for x+y=nz: the least k such that every 2-coloring of {1,…,k} has a monochromatic distinct-variable solution | Ramsey theory |
| A394661 | Triangle T(n,k): number of prime knots with n crossings and three-genus k | Knot theory |
| A395521 | Number of non-isomorphic abelian groups appearing as the sandpile group K(G) over graphs on n vertices | Algebraic graph theory |
| A395644 | Number of fibered prime knots with n crossings | Knot theory |
| A396815 | Largest subset of the cyclic group Z/nZ that splits into two sum-free sets | Additive combinatorics |
| A396816 | Largest subset of Z/nZ that splits into three sum-free sets | Additive combinatorics |
| A397074 | NPN-equivalence classes of n-variable Boolean functions whose sensitivity equals n | Boolean complexity |
| A397075 | NPN classes of n-variable Boolean functions whose (real) polynomial degree equals n | Boolean complexity |
| A397076 | NPN classes of n-variable Boolean functions whose decision-tree complexity equals n | Boolean complexity |
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 verified by a second solver; a reviewer’s observation on A396815 turned into a small theorem, that its odd-indexed values are exactly twice the maximal sum-free set size (A211316).
A397074, A397075, and A397076 come from a complete census of all 616,126 NPN-equivalence classes of Boolean functions on up to five variables. 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. The counts were re-derived by an independent clean-room recomputation anchored to the known class totals (A000370).
Extended sequences
Classical Zarankiewicz-problem sequences (originally by N. J. A. Sloane) where I computed and added new terms past the known frontier.
| Sequence | Description | My terms |
|---|---|---|
| A006615 | z(n,n;3,4): least k forcing an all-ones 3x4 submatrix in an n x n 0/1 matrix | a(10)=67; a(11)=79 |
| A006622 | z(n,n+1;3,4): same for n x (n+1) matrices | a(9)=61; a(10)=73; a(11)=85 |
| A006625 | z(n,n+2;3,4): same for n x (n+2) matrices | a(9)=67; a(10)=79 |
The Zarankiewicz extensions use SAT: a satisfying assignment exhibits a dense matrix (a lower bound), and an unsatisfiable instance proves the matching upper bound. The most recent terms required 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 1 2026. They carry the method into combinatorial game theory, Sprague-Grundy values and Sylver coinage, alongside one more Ramsey number.
| Sequence | Description | My term | Domain |
|---|---|---|---|
| A006672 | Ramsey number r(C_4, K_{1,n}) | a(11)=16 | Ramsey theory |
| A316632 | Sprague-Grundy value of Node-Kayles on the 3 x n grid graph | a(17)=2 | Combinatorial game theory |
| A046671 | Nim-values G(3,n) for Sylver coinage | a(17)=14 | Combinatorial game theory |
Each new term was reproduced against every published term of its sequence by an independent from-scratch program before submission. 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.
Recently approved
- A006672 a(11)=16 (Ramsey r(C_4, K_{1,n})), A316632 a(17)=2 (Node-Kayles on the 3 x n grid), and A046671 a(17)=14 (Sylver coinage G(3,n)) were published July 1 2026.
- All seven new Zarankiewicz terms above (A006615 a(11)=79; A006622 a(10)=73, a(11)=85; A006625 a(10)=79) are now published.
- A396815 and A396816 (the sum-free partition sequences) were approved June 15 2026.
- A397074, A397075, and A397076 (the Boolean-measure NPN-class sequences) were approved June 15 2026.
The A394445 b-file (500 terms) is approved and live.
Reviewers
These submissions were reviewed and approved by OEIS editors including Michel Marcus, Sean A. Irvine, Max Alekseyev, and Jon E. Schoenfield.
Reproduce
Every term here is reproducible. The
portfolio repository keeps a single source
of truth for the contributions and, for each sequence, a standalone
reproduce.py that recomputes its values from scratch and checks them against
the published terms before anything is submitted.
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.