A253786 a(3n) = 0, a(3n+1) = 0, a(3n+2) = 1 + a(n+1).

0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 5

Offset: 0

Author

Antti Karttunen, Jan 22 2015

Keywords

• nonn
• easy

Comments

For n >= 1, a(n) gives the distance of n in square array A191450 from its leftmost column.

The sequence 0,1,0,0,0,2,0,...,i.e., (a(n)) with the first term removed, is the unique fixed point of the constant length 3 morphism N -> 0 N+1 0 on the infinite alphabet {0,1,...,N,...}. - Michel Dekking, Sep 09 2022

a(n) is the number of trailing 1 digits of n-1 written in ternary, for n>=1. - Kevin Ryde, Sep 09 2022

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10000

Crossrefs

One less than A254046.

Cf. A007814, A007949, A048673, A064216, A135523, A191450 (A254051), A253887, A253894, A254045, A254103, A254104, A245611.

Programs

Mathematica

With[{nmax=200},IntegerExponent[2Range[0,nmax]-1,3]] (* Paolo Xausa, Nov 09 2023 *)

PARI

a(n) = n--; my(ret=0,r); while([n,r]=divrem(n,3); r==1, ret++); ret; \\ Kevin Ryde, Sep 13 2022

Formula

Other identities and observations. For all n >= 1:

a(n) = A254046(n)-1.

a(n) <= A254045(n) <= A253894(n).

a(3n-1) = A254046(n). - Cyril Damamme, Aug 04 2015

a(n) = A007949(2n-1), i.e., the 3-adic valuation of 2n-1. - Cyril Damamme, Aug 04 2015

From Antti Karttunen, Sep 12 2017: (Start)

For all n >= 1:

a(n) = A007814(A064216(n)) = A007814(A254104(n)) = A135523(A245611(n)).

a(A048673(n)) = a(A254103(n)) = A007814(n).

a(A244154(n)) = A007814(1+n).

a(A245612(n)) = A135523(n). (End)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Nov 16 2023