The OEIS meta sequence and subway stations || Math ∩ Programming
The OEIS meta sequence and subway stations<br>#shortform<br>#depths of OEIS<br>#oeis<br>2026-04-09<br>A051070 is a sequence about OEIS sequences.<br>a(n) is the n-th term in sequence A_n (or -1 if A_n doesn’t have enough<br>terms).<br>So the first term in A051070 is 1 because A000001 is the number of<br>groups of order n, and that sequence has 1 as its entry in index 1. A000002 is<br>the Kolakoski sequence (what? For another time) and has value 2 in entry 2.<br>The sequence continues: 1, 2, 1, 0, 2, 3, 0, 7, 8, 4, 63, 1, 316, …<br>At first you might think, “what in the Gödel?” What if the arbitrary indexing<br>of the OEIS changes over time? Aren’t these sequences supposed to be defined by<br>mathematical rules?<br>Not the fun ones, apparently. In the comments, Pontus von Brömssen noted<br>that a(58) has 58669977298272603 digits, so it’s too large to include<br>in the database entry for A051070. a(66) is the first unknown value,<br>because A000066 (Smallest number of vertices in trivalent graph with girth (shortest cycle) = n)<br>is only known up to 12 vertices.<br>And then we get to my two favorite quirks about this sequence.<br>The first is that the first time a(n) = -1 occurs, it’s for n = 53 and 54,<br>quoting the OEIS, “in both cases because the relevant New York subway lines do not have enough stops.”<br>What? Why are New York Subway lines involved?<br>Turns out, the OEIS has roughly a dozen sequences of numbered stops on train lines.<br>A000053 is “Local stops on New York City 1 Train (Broadway-7 Avenue Local) subway.”<br>A001049 is “Numbered stops in Manhattan on the Lexington Avenue subway.”<br>Of course, this chips away even further at the idea that OEIS sequences<br>need to have a mathematical definition removed from worldly messiness.<br>Digging around, I could only find a short note in this Numberphile video<br>where Neil Sloane (who created OEIS and added these entries)<br>mentioned that they’re commonly used on math quizzes and tests.<br>If you know someone who has used train lines on their quizzes,<br>and didn’t already know about these OEIS entries,<br>please let me know. I need this to be a common organic experience.<br>The second quirk is that A051070 leaves open the question of<br>what the value of a(51070) is.<br>It gets worse with A102288, which is defined as<br>1 + the n-th term in sequence A_n. (There are some slight differences about offsets here, but I’m using A102288 because it has juicier comments)<br>Even if there was a default value for the 102288-th entry in this sequence,<br>it would contradict its own definition.<br>There is an argument in the comments section, which starts with an unattributed “What is a(102288)?!”<br>M. F. Hasler complained in 2017: “The term a(102288) has no possible value according to the present definition, so the definition of this term should be changed.”<br>Neil Sloane replied the same day: “I disagree with the previous comment! I prefer the present, deliberately paradoxical, definition.”<br>In the age-old battle between whimsy and well-definedness, whimsy wins again.<br>Want to respond? Send me an email,<br>post a webmention,<br>or find me elsewhere on the internet.<br>This article is syndicated on:<br>Mastodon<br>Bluesky
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