A214848 - OEIS

A214848

First difference of A022846.

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

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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

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

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.

S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, on ResearchGate.

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 *)