A082850 Let S(0) = {}, S(n) = {S(n-1), S(n-1), n}; sequence gives S(infinity).

1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 5, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 5, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1

Offset: 1

Author

Benoit Cloitre, Apr 14 2003

Keywords

• nonn
• hear

Comments

Sequence counts up to successive values of A001511; i.e., apply the morphism k -> 1,2,...,k to A001511. If all 1's are removed from the sequence, the resulting sequence b has b(n) = a(n)+1. A101925 lists the positions of 1's in this sequence.

The geometric mean of this sequence approaches the Somos constant (A112302). - Jwalin Bhatt, Jan 30 2025

Examples

			S(1) = {1}, S(2) = {1,1,2}, S(3) = {1,1,2,1,1,2,3}, etc.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..65535

Crossrefs

Cf. A082851 (partial sums).

Cf. A001511, A101925, A112302.

Cf. A215020.

Programs

Mathematica

Fold[Flatten[{#1, #1, #2}] &, {}, Range[5]] (* Birkas Gyorgy, Apr 13 2011 *)
Flatten[Table[Length@Last@Split@IntegerDigits[2 n, 2], {n, 20}] /. {n_ ->Range[n]}] (* Birkas Gyorgy, Apr 13 2011 *)

Python

S = []; [S.extend(S + [n]) for n in range(1, 8)]
print(S) # Michael S. Branicky, Jul 02 2022

Python

from itertools import count, islice
def A082850_gen(): # generator of terms
    S = []
    for n in count(1):
        yield from (m:=S+[n])
        S += m #
A082850_list = list(islice(A082850_gen(),20)) # Chai Wah Wu, Mar 06 2023

Formula

a(2^m - 1) = m.

If n = 2^m - 1 + k with 0 < k < 2^m, then a(n) = a(k). - Franklin T. Adams-Watters, Aug 16 2006

a(n) = log_2(A182105(n)) + 1. - Laurent Orseau, Jun 18 2019

a(n) = 1 + A215020(n). - Joerg Arndt, Mar 04 2025