A356994 a(n) = n - b(b(b(n))), where b(n) = A356988(n).

0, 1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 10, 10, 11, 12, 13, 14, 15, 16, 16, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 26, 26, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 42, 42, 42, 42, 42, 42, 42, 43, 44, 45

Offset: 1

Author

Peter Bala, Sep 09 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 2, 4, 6, 10, 16, 26, 42, 68, 110, ..., the sequence {2*Fibonacci(k): k >= 2}

The plateau of height 2*F(k), k >= 2, has length equal to Fibonacci(k-2), starting at abscissa value n = Fibonacci(k+2) and ending at abscissa n = 3*Fibonacci(k).

Crossrefs

Cf. A000045, A356988, A356991, A356992, A356993, A356995, A356996, A356997, A356998, A356999.

Programs

Maple

#  b(n) = A356988
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(b(b(n))), n = 1..100);

Formula

The sequence is determined by the initial values a(1) = 0, a(2) = 1 and the pair of formulas

1) a(n) = 2*Fibonacci(k) for n in the integer interval [Fibonacci(k+2), 3*Fibonacci(k)], k >= 2, and

2) for k >= 2, a(3*Fibonacci(k) + j) = 2*Fibonacci(k) + j for 0 <= j <= 2*Fibonacci(k-1).