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A214848 First difference of A022846. 5
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, 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, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of triangular numbers in interval [n^2, (n+1)^2).
From Michel Dekking, Sep 20 2022: (Start)
(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).
(a(n)) is the fixed point of the morphism 1->12121, 2->1212121.
This is proved by writing the 0-1 version psi: 0->01010, 1->0101010 of this morphism as a composition
psi = psi_1 psi_3 psi_1 psi_4,
where the psi_i are the three elementary Sturmian morphisms
psi_1: 0->01, 1->0, psi_3: 0->0, 1->01, psi_4: 0->0, 1->10.
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)
REFERENCES
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.
LINKS
Svetlana Jitomirskaya, Small denominators and multiplicative Jensen's formula, ICM 2022. See the initial slides "Playing with numbers".
M. Lothaire, Algebraic combinatorics on words, Cambridge University Press. Online publication date: April 2013; Print publication year: 2002.
FORMULA
For n > 0: a(n) = A006338(n). - Reinhard Zumkeller, Mar 03 2014
EXAMPLE
28 is in [25, 36), a(5) = 1.
36 and 45 are in [36, 49), a(6) = 2.
MATHEMATICA
Differences[Round[Sqrt[2]Range[0, 100]]] (* Harvey P. Dale, Jun 14 2020 *)
PROG
(Haskell)
a214848 n = a214848_list !! n
a214848_list = zipWith (-) (tail a022846_list) a022846_list
-- Reinhard Zumkeller, Mar 03 2014
CROSSREFS
Sequence in context: A175922 A214856 A006337 * A006338 A020903 A133083
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Mar 08 2013
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)