This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A214848 #34 Sep 20 2022 11:09:54 %S A214848 1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1, %T A214848 1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1, %U A214848 2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1 %N A214848 First difference of A022846. %C A214848 Number of triangular numbers in interval [n^2, (n+1)^2). %C A214848 From _Michel Dekking_, Sep 20 2022: (Start) %C A214848 (a(n)) is an inhomogeneous Sturmian sequence s(alpha, rho) with slope alpha = sqrt(2) and intercept 1/2, since A022846(n) = floor(n*sqrt(2) + 1/2). %C A214848 (a(n)) is the fixed point of the morphism 1->12121, 2->1212121. %C A214848 This is proved by writing the 0-1 version psi: 0->01010, 1->0101010 of this morphism as a composition %C A214848 psi = psi_1 psi_3 psi_1 psi_4, %C A214848 where the psi_i are the three elementary Sturmian morphisms %C A214848 psi_1: 0->01, 1->0, psi_3: 0->0, 1->01, psi_4: 0->0, 1->10. %C A214848 By Lemma 2.2.18 in Lothaire it then follows that the 0-1 word (a(n)-1) = A214848 is fixed by the morphism psi (note that in Lothaire psi_1 is phi, psi_3 is G, and psi_4 is G^~). (End) %D A214848 S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, in Number theory and its applications (Kyoto, 1997), pp. 347-373, Kluwer Acad. Publ., Dordrecht, 1999. %H A214848 Reinhard Zumkeller, <a href="/A214848/b214848.txt">Table of n, a(n) for n = 0..10000</a> %H A214848 Svetlana Jitomirskaya, <a href="https://www.youtube.com/watch?v=N7uktgo1vM4">Small denominators and multiplicative Jensen's formula</a>, ICM 2022. See the initial slides "Playing with numbers". %H A214848 M. Lothaire, <a href="http://tomlr.free.fr/Math%E9matiques/Fichiers%20Claude/Auteurs/aaaDivers/Lothaire%20-%20Algebraic%20Combinatorics%20On%20Words.pdf">Algebraic combinatorics on words</a>, Cambridge University Press. Online publication date: April 2013; Print publication year: 2002. %H A214848 S.-I. Yasutomi, <a href="https://www.researchgate.net/publication/268247171_On_Sturmian_sequences_which_are_invariant_under_some_substitutions">On Sturmian sequences which are invariant under some substitutions</a>, on ResearchGate. %F A214848 For n > 0: a(n) = A006338(n). - _Reinhard Zumkeller_, Mar 03 2014 %e A214848 28 is in [25, 36), a(5) = 1. %e A214848 36 and 45 are in [36, 49), a(6) = 2. %t A214848 Differences[Round[Sqrt[2]Range[0,100]]] (* _Harvey P. Dale_, Jun 14 2020 *) %o A214848 (Haskell) %o A214848 a214848 n = a214848_list !! n %o A214848 a214848_list = zipWith (-) (tail a022846_list) a022846_list %o A214848 -- _Reinhard Zumkeller_, Mar 03 2014 %Y A214848 Cf. A022846, A006337, A006338. %K A214848 nonn,easy %O A214848 0,2 %A A214848 _Philippe Deléham_, Mar 08 2013