A356989 a(n) = n - a^[3](n - a^[4](n-1)) with a(1) = 1, where a^[3](n) = a(a(a(n))) and a^[4](n) = a(a(a(a(n)))).

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

Offset: 1

Author

Peter Bala, Sep 08 2022

Keywords

• nonn
• easy

Comments

This is the third 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](s(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. The present sequence is the case k = 3. For other cases see A006165 (k = 1), A356988 (k = 2) and A356990 (k = 4).

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 4, 6, 9, 13, 19, 28, 41, 60, ..., conjecturally A000930.

The plateaus start at absiccsa values n = 5, 8, 12, 17, 25, 37, 54, 79, ..., conjecturally A179070, and terminate at abscissa values n = 6, 9, 13, 19, 28, 41, 60, ..., conjecturally A000930.

Crossrefs

Cf. A000930, A006165, A179070, A356988, A356990.

Programs

Maple

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