A092400 Fixed point of the morphism 1 -> 1121211, 2 -> 1121212121211, starting from a(1) = 1.

1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Author

Philippe Deléham, Mar 21 2004

Keywords

• easy
• nonn

Comments

Length of n-th run of identical symbols in A051069.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are fixed points of mappings.

Crossrefs

Cf. A007417, A051069.

Programs

Mathematica

Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 1, 2, 1, 2, 1, 1}, 2 -> {1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1}})]}], {1}, 3] (* Robert G. Wilson v, Feb 26 2005 *)

Python

from sympy import integer_log
def A007417(n):
    def f(x): return n+x-sum(((m:=x//9**i)-2)//3+(m-1)//3+2 for i in range(integer_log(x,9)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m
def A092400(n): return A007417(n)-A007417(n-1) if n>1 else 1 # Chai Wah Wu, Feb 16 2025

Formula

Sum_{k=1..n} a(k) = A007417(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/3. - Amiram Eldar, Apr 11 2025

Extensions

More terms from Robert G. Wilson v, Feb 26 2005