Checking 213 OEIS recurrence conjectures against their b-files finds one failure, already known

Many entries in the On-Line Encyclopedia of Integer Sequences carry a line such as "Conjecture: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3)". Most such lines were found by fitting the terms known at the time, and the b-file stored beside the entry often lists hundreds or thousands more. We asked how often these conjectures fail on their own b-files and whether the failures are data errors or counterexamples. From the public OEIS data, 23,883 entries mention a conjecture and 1,179 have a line our parser accepts; we drew a random 500 and tested every line that its b-file extends by at least 30 terms. One of 213 tested conjecture lines failed on its b-file, 0.47 percent, with a 95 percent Wilson interval of 0.08 to 2.6 percent.c1 The one failure, A031740, holds for 64 indices and fails at n = 65, and the entry has recorded the conjecture as false since 2016, so the sweep found no new counterexample.c2 Planted errors were caught in 212 of 212 passing lines, and equations applied to other entries' terms passed at a mean rate of 0.005.c3 The sampling frame changed after the preregistered route failed, and the change is recorded. The result says the check is cheap and, on this kind of line, low in yield.

The OEISDefinition: OEIS mathematics (reference databases) The On-Line Encyclopedia of Integer Sequences, a public database of number sequences in which each entry carries the first terms, formulas, comments and links contributed by many people. Excludes a peer-reviewed journal; a line in an entry is a contribution, and a conjecture line in it carries no proof. full definition is a public database of integer sequences, online since 1995 and holding over 300,000 entries when Sloane described its recent history in 2018 [2]. Each entry lists the first terms of a sequence in a short data section, with formulas, comments and references contributed by many people. Beside most entries sits a b-fileDefinition: b-file the OEIS A text file stored beside an OEIS entry that lists the sequence's index and term pairs much further than the entry's own data section, often to thousands of terms. Excludes the entry's data section, which holds only the first few dozen terms. full definition, a plain list of index and term pairs that often runs to thousands of terms.

Some entries carry a conjecture lineDefinition: conjecture line the OEIS, our usage A comment or formula line in an OEIS entry that states an equation for the sequence and labels it a conjecture. Most were found by fitting the terms then known. Excludes a line marked as proved, and a statement with no equation. full definition: an equation for the sequence, offered without proof. Many are a linear recurrenceDefinition: linear recurrence mathematics (combinatorics) An equation that gives each term of a sequence as a fixed weighted sum of a set number of earlier terms, possibly with terms in n. Here with integer coefficients. Excludes an equation with a product of earlier terms, and an equation with coefficients that are not integers. full definition with integer coefficients, found by fitting the terms then listed. Fitting of this kind is a standard method in computer algebra. Kauers and Koutschan [3] describe guessing a recurrence from finitely many terms by linear algebra, note that the overdetermination of the linear system is one source of trust in a guess, and show how to guess from fewer terms by restricting the coefficients. In a later search across the OEIS [4] they found conjectured recurrences for sequences not previously known to satisfy one, proved some, and showed that a recurrence exists for others without proving the guessed one.

Our question was smaller. The b-files already hold the data a mechanical check needs. We asked what share of conjecture lines fail on their own b-files, and whether each failure is a data error or a true counterexampleDefinition: counterexample mathematics (logic) A single case where a general statement is false. One is enough to refute a conjecture. Excludes a term that disagrees because the data are wrong. full definition. The decision it informed was whether such lines are worth a routine check as a cheap source of counterexamples, and how to read one when it appears in our own work. Our searches found the guessing literature above and no published sweep of this exact kind; that is what the searches returned, and it does not show that none exists.

The preregistrationDefinition: preregistration research method (metascience) A dated written plan for a study, fixed and hash-locked before any result is seen: the question, the data, the statistic, the controls, the forecasts and the kill conditions. Excludes a plan revised after looking at the outcome, and a protocol written only for the record after the run. full definition was written and hash-locked on 31 October 2026, before any entry was sampled beyond one probe request that returned an unrelated entry. It names the sampling route, the parser rules, the definition of tested and of failed, the planted errors, the control, the forecasts and the kill conditions.

Kill conditions, without discretion:

Each kill conditionDefinition: kill condition research method (our usage, after Popper's falsification) A rule written in the preregistration that, when met, decides against the claim with no room for judgement after the run. Excludes reasons found after the run for setting a result aside. full definition and forecast became a rule set whose pass or fail is a verdictDefinition: verdict research method (our usage) The pass or fail of one preregistered rule set against the run's numbers, read off by code. Excludes an interpretation written after reading the numbers. full definition read off by code.

The locked plan sampled entries through the OEIS search pages at random offsets. That route returned only 10 entries, because the site asks for a sign-in past the first 100 search results, and we did not work around it. The first run tested 2 lines and was void under the first kill condition. Having seen only those 2 lines, both passing, we changed the sampling frameDefinition: sampling frame survey statistics The list from which a sample is drawn. A result describes the frame, so a change of frame changes what the result is about. Excludes the sample itself. full definition: the public OEIS data repository, every entry with a parsable conjecture line, then a seeded random 500. The change is recorded as a deviationDefinition: deviation research method (metascience) A dated, recorded change from the preregistered plan, stated with its reason and with what had been seen when it was made. Excludes a change made silently, and a change made after seeing the outcome it would affect without saying so. full definition with its reason and what had been seen. The new frame is wider than the locked one, and the second run used the same locked plan, rules and forecasts.

The data are the OEIS entries from the public oeis/oeisdata repository and the b-files from oeis.org [1], retrieved on 31 October 2026. OEIS content is licensed CC BY-SA 4.0 under the OEIS End-User License Agreement. We used it as data for checks; this paper reports counts and quotes one entry's conjecture and comment lines with attribution.

Of all entries, 23,883 mention a conjecture and 1,179 have a conjecture line our parser accepts. The seeded random 500 entries held 531 lines that mention a conjecture and a term a(n), and 512 of those parsed. Only about 5 percent of the entries that mention a conjecture had a line our parser accepts: 1,179 of 23,883.c4 The result therefore describes simple linear recurrences, not OEIS conjectures in general.

A line was parsed only if it states an equation in $n$ and earlier terms $a(n-k)$ with integer coefficients. A pattern match finds the candidate equation, and a parser whitelistDefinition: parser whitelist computer science (parsing) A rule that turns each line into a syntax tree and accepts it only if every part is on a fixed list: integers, n, the terms a(n-k), addition, subtraction, multiplication and powers. Excludes any line that needs division, sums, named functions or words to state. full definition accepts it only if every part of its syntax tree is an integer, $n$, a term $a(n-k)$, addition, subtraction, multiplication or a power. Anything else was skipped and counted.

For each sampled entry, the first conjecture line that parsed and qualified was tested; at most one line per entry was tested. A line qualified when the b-file holds at least 30 more terms than the entry's data section and the equation can be evaluated at 10 or more indices. The equation was evaluated at every index in the b-file's range where all the terms it needs exist. A line fails if its two sides differ at any such index; the first failing index and the number of indices it held for were recorded. This left 213 tested lines, with a median of 967 b-file terms beyond the data section.

With $k$ failures among $N$ tested lines, the failure rate is $f=k/N$. Its Wilson intervalDefinition: Wilson interval statistics A 95 percent interval for a proportion, found by inverting the score test. It stays inside 0 to 1 and behaves well when the count is small or near zero. Excludes the simple interval of the proportion plus or minus 1.96 standard errors, which fails near zero. full definition [5] at $z$ = 1.96 has centre $c$ and half-width $w$:

The planted checkDefinition: planted check research method; called a positive control in experimental science Data built so that the right answer is known, run through the same analysis before the real data, to show that the analysis can see the effect when it is there. Excludes any check made on the real data, which cannot show what the analysis misses. full definition raised the first integer coefficient of each passing equation by one and ran the check again. A planted line should fail. Of the 213 tested lines, 212 passed, so 212 planted versions were built. All 212 were caught.

The cross-entry controlDefinition: control statistics and experimental design A version of the data with the structure under test removed (for example shuffled or pooled), scored the same way as the real data, so the real score has something to beat. Excludes a comparison with a different dataset or with a published number. full definition applied each equation to a different, randomly chosen entry's terms, up to index 200. One draw takes 30 equations at random, applies each to another entry's terms, and records the share that pass. This share is the pass rate of an equation on a sequence it was not written for, the chance level of the test. Over 200 draws the mean share was 0.0048, and the 95th percentile of this null distributionDefinition: null distribution statistics The spread of scores a control gives over many random draws. Its 95th percentile is the bar a real score must clear. Excludes a theoretical distribution assumed rather than drawn from the data. full definition was 0.033, one equation in 30.

The real pass rate, 0.995, was above every draw. The p value from drawsDefinition: p value (from draws) statistics The share of control draws scoring at or above the real score, with one added to the count and to the number of draws, so it is never zero. Excludes a p value from a textbook formula that assumes a distribution. full definition is

with $k$ the number of draws at or above the real pass rate and $n$ the number of draws. Here $k$ = 0 and $n$ = 200, so $p$ = 0.005, the smallest value 200 draws can give.

Figure 1 follows the counts from the whole corpus to the one failure.

One of 213 tested conjecture lines failed on its b-file, 0.47 percent, with a 95 percent Wilson interval of 0.08 to 2.6 percent.c1 Figure 2 places the rate against the forecast of 4 percent and the preregistered 1 percent rule.

The failing line is in entry A031740. Its conjecture, contributed by Colin Barker on 30 September 2012, reads:

The entry's data section has 32 terms and its b-file 10,000. The one failure, A031740, holds for 64 indices and fails at n = 65, and the entry has recorded the conjecture as false since 2016, so the sweep found no new counterexample.c2 The stored copy of the entry carries a comment by Chai Wah Wu dated 19 June 2016: the conjecture is false, and 3940288 is a term not of the form $m(961m+1)$. The preregistration asked for a second code path before a failure counts; the result file records no such rerun. The check by hand was reading this comment, which confirms the failure is a true counterexample and not a b-file error.

Table 1 gives the counts and rates.

Table 2 sets the forecasts written before the run against the outcomes.

The sweep shows that a mechanical check of OEIS conjecture lines is cheap: the analysis took about 6 seconds once the data were stored. On simple linear recurrences it is also low in yield. The lines held at a rate of 99.5 percent, the failure rate came in an order of magnitude under the forecast, and the single failure had been caught by a contributor four years after it was posted.

Two features of the lines explain much of this. They were fitted to data, so passing on further terms is what a correct fit does. And a recurrence of order $d$ fitted to a few more than $d$ terms is close to determined by them, so the guess is either right or wrong in a way that shows quickly; A031740's conjecture failed just past twice the length of its data section. Kauers and Koutschan [3] point to the same property, overdetermination, as a source of trust in a guessed recurrence. Our pass rate is a measurement of how well that trust held on one class of OEIS line.

The more interesting conjectures are the ones our parser skipped: 95 percent of the entries that mention a conjecture state something our whitelist cannot read; we did not classify what those lines say. A sweep of those would need a different checker for each kind of line.