A356992 Then a(n) = n - b(n - b(n - b(n - b(n - b(n - b(n)))))) for n >= 2, where b(n) = A356988(n).

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

Offset: 2

Author

Peter Bala, Sep 08 2022

Keywords

• nonn
• easy

Comments

The sequence is slow, that is, for n >= 2, a(n+1) - a(n) is either 0 or 1. The sequence is unbounded.

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, 7, 11, 18, 29, 47, ..., conjecturally the Lucas sequence {A000032(k): k >= 3}.

The plateaus start at abscissa values n = 5, 10, 16, 26, 42, 68, .... Apart from the first term 5, this appears to be the sequence {2*Fibonacci(k): k >= 5}.

The plateaus end at abscissa values n = 7, 12, 19, 31, 50, 81, ..., conjecturally the sequence {A013655(k): k >= 3}.

The sequence of plateau lengths begins 2, 2, 3, 5, 8, 13, .... Apart from the first term 2, this appears to be the sequence {Fibonacci(k): k >= 3}.

The slow sequences {a(a(n)): n >= 3} and {a(a(a(n))): n >= 4} appear to have similar properties to the present sequence. The slow sequence {n - a(n): n >= 2} appears to have plateaus at heights given by the Fibonacci sequence. See the Example section.

Examples

			Related sequences:
1) {n - a(n): n >= 2}
  1, 1, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 35, 36, 37, ...
The line graph of the sequence has plateaus at heights 3, 5, 8, 13, 21, 34, ..., conjecturally the Fibonacci numbers A000045.
2) {a(a(n)): n >= 3}
  1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 47, 47, 47, 47, ...
The line graph of the sequence has plateaus at heights 3, 4, 7, 11, 28, 29, ..., conjecturally the Lucas numbers A000045.
3) {a(a(a(n))): n >= 4}
  1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29, ...
The line graph of the sequence has plateaus at heights (2), 3, 4, 7, 11, 28, 29, ..., conjecturally the Lucas numbers A000045.
		

Crossrefs

Cf. A000032, A000045, A013655, A356988, A356991 - A356999.

Programs

Maple

b := proc(n) option remember; if n = 1 then 1 else n - b(b(n - b(b(b(n-1))))) end if; end proc:
seq(n - b(n - b(n - b(n - b(n - b(n - b(n)))))), n = 2..100);