A356990 a(n) = n - a^[4](n - a^[5](n-1)) with a(1) = 1, where a^[4](n) = a(a(a(a(n)))) and a^[5](n) = a(a(a(a(a(n))))).

1, 1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 19, 19, 20, 21, 22, 23, 24, 25, 26, 26, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 36, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 50, 50, 50, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 69, 69, 69, 69, 69, 69, 69, 70, 71

Offset: 1

Author

Peter Bala, Sep 08 2022

Keywords

• nonn
• easy

Comments

This is the fourth sequence in a family of nested-recurrent sequences with apparently similar structure defined as follows. Given a sequence s = {s(n); n >= 1} we define the k-th iterated sequence s^[k] by putting s^[1](n) = s(n) and setting s^[k](n) = s^[k-1](u(n)) for k >= 2. For k >= 1, we define a nested-recurrent sequence {u(n): n >= 1}, dependent on k, by putting u(1) = 1 and setting u(n) = n - u^[k](n - u^[k+1](n-1)) for n >= 2. This is the case k = 4. For other cases see A006165 (k = 1), A356988 (k = 2) and A356989 (k = 3).

The sequence is slow, that is, for n >= 1, a(n+1) - a(n) is either 0 or 1.

The line graph of the sequence {a(n)} thus consists of a series of plateaus (where the value of the ordinate a(n) remains constant as n increases) joined by lines of slope 1.

The sequence of plateau heights begins 5, 7, 10, 14, 19, 26, 36, 50, ..., which appears to be A003269.

The plateaus start at absiccsa values n = 6, 9, 13, 18, 24, 33, 46, 64, ..., which appears to be A014101, and terminate at abscissa values 7, 10, 14, 19, 26, 36, 50, ..., conjecturally A003269.

Crossrefs

Cf. A003269, A006165, A014101, A356988, A356989.

Programs

Maple

a := proc(n) option remember; if n = 1 then 1 else n - a(a(a(a(n - a(a(a(a(a(n-1))))))))) end if; end proc:
seq(a(n), n = 1..100);

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A356990(n): return 1 if n <= 1 else n-A356990(A356990(A356990(A356990(n-A356990(A356990(A356990(A356990(A356990(n-1))))))))) # Chai Wah Wu, Oct 01 2022