2026-05-09T03:15:32Z https://yabu.me Peter Luschny https://yabu.me/npub1ky725xgartjzwvtuqrgrpc0z2d54cf88nm3swawurrmpdezaq0lqr46pll https://media.mathstodon.xyz/accounts/avatars/109/269/476/832/714/186/original/a509de00d5b1aa7b.jpg https://media.mathstodon.xyz/accounts/avatars/109/269/476/832/714/186/original/a509de00d5b1aa7b.jpg https://yabu.me/nevent1qqszrksdxxa5pw946eq4403khu6h398nf6752gsm5pneemfmckjna9szyzcne2ser5dwgfe30sqdqv8puffkjhpyu70wxpm4msv0v9hyt5pludtlhe9 In reply to <a href='/nevent1qqsqhrtt2c8j6sqf6k9h6ug3qrg083ryd33phufxr0dl04r4as0m26csah7p3'>nevent1q…h7p3</a><br/>_________________________<br/><br/>The reference to the Eneström Number (Euler&#39;s Werkeverzeichnis) reveals the answer.<br/><br/>Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, volume 1.<br/> Leonhard Euler<br/><br/> On the transformation of series<br/><br/>Igor Pak, History of Catalan numbers, writes:<br/> <br/>&#34;On September 4, 1751, Euler wrote a letter to Goldbach which among other things included his discovery of Catalan numbers. Euler was in Berlin at that time, while his friend and former mentor Goldbach was in St. Petersburg. Euler defines Catalan numbers C_n as the number of triangulations of (n + 2)-gon.&#34;<br/><br/>Anyone interested in the history of these numbers should definitely read Pak&#39;s excellent essay.<br/><br/><a href="https://scholarlycommons.pacific.edu/euler-works/212/">https://scholarlycommons.pacific.edu/euler-works/212/</a><br/><br/><a href="https://www.math.ucla.edu/~pak/papers/cathist4.pdf">https://www.math.ucla.edu/~pak/papers/cathist4.pdf</a><br/><br/>A very good website:<br/><a href="https://www.math.ucla.edu/~pak/lectures/Cat/pakcat.htm">https://www.math.ucla.edu/~pak/lectures/Cat/pakcat.htm</a><br/><br/><a href="https://arxiv.org/pdf/1905.10438">https://arxiv.org/pdf/1905.10438</a> 2025-12-10T06:15:19Z https://yabu.me/nevent1qqsffuc3hvum2xw6hqje0lc28m4r6yfu8kt3grdgr80czpuq3yqqftgzyzcne2ser5dwgfe30sqdqv8puffkjhpyu70wxpm4msv0v9hyt5plur4nymv Written in 1755. Guess by whom. <br/><br/>Anyone who fails to recognize the numbers will be excluded from the OEIS next year.<br/><br/>Eneström-Number E212, § 13.<br/> <img src="https://media.mathstodon.xyz/media_attachments/files/115/691/928/853/685/900/original/2f75d9a6866e1b77.png"> <br/> 2025-12-09T22:19:09Z https://yabu.me/nevent1qqs9z0jp5wlpz35gdfrg46m6c6k92j4gfcu284yhxe0fn7fdkfraarszyzcne2ser5dwgfe30sqdqv8puffkjhpyu70wxpm4msv0v9hyt5plumv7ha5 In reply to <a href='/nevent1qqsrvdnjg2mvueth7dqe770g9649x4ntl9md9r5msvst03xn2veugtqak4vum'>nevent1q…4vum</a><br/>_________________________<br/><br/>Thanks for reminding us of this lovely entry. <br/><br/>Officially, at least three, preferably four, terms are required. 2025-09-02T12:06:44Z https://yabu.me/nevent1qqs9nw79kkt5yagn6esp902p3lhsqrmz3z9wlmudjd387uyj5vafluczyzcne2ser5dwgfe30sqdqv8puffkjhpyu70wxpm4msv0v9hyt5plu8rf275 Like old Julius famously said: <br/><br/>&#34;Omnes numeri naturales in partes tres divisi sunt.&#34;<br/><br/>Translated into OEIS parlance, this means:<br/><br/>A001477 = A000040 U A002808 U ? <br/>where &#39;U&#39; denotes the disjoint union.<br/><br/>The problem is that the OEIS is a database for _infinite_ sequences. And so this fundamental relationship could not be expressed in terms of OEIS-numbers. <br/><br/>Only rarely do the editors make an exception and allow finite sequences. Yesterday they were generous, and now the gap has been closed.<br/><br/>But what is &#39;?&#39; anyway? 2025-09-02T11:27:30Z