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%I #34 Sep 20 2022 11:09:54
%S 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 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 2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1
%N First difference of A022846.
%C Number of triangular numbers in interval [n^2, (n+1)^2).
%C From _Michel Dekking_, Sep 20 2022: (Start)
%C (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 (a(n)) is the fixed point of the morphism 1->12121, 2->1212121.
%C This is proved by writing the 0-1 version psi: 0->01010, 1->0101010 of this morphism as a composition
%C psi = psi_1 psi_3 psi_1 psi_4,
%C where the psi_i are the three elementary Sturmian morphisms
%C psi_1: 0->01, 1->0, psi_3: 0->0, 1->01, psi_4: 0->0, 1->10.
%C 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 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 Reinhard Zumkeller, <a href="/A214848/b214848.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 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 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 For n > 0: a(n) = A006338(n). - _Reinhard Zumkeller_, Mar 03 2014
%e 28 is in [25, 36), a(5) = 1.
%e 36 and 45 are in [36, 49), a(6) = 2.
%t Differences[Round[Sqrt[2]Range[0,100]]] (* _Harvey P. Dale_, Jun 14 2020 *)
%o (Haskell)
%o a214848 n = a214848_list !! n
%o a214848_list = zipWith (-) (tail a022846_list) a022846_list
%o -- _Reinhard Zumkeller_, Mar 03 2014
%Y Cf. A022846, A006337, A006338.
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Mar 08 2013
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