A356993 a(n) = b(n - b(n - b(n - b(n)))) for n >= 2, where b(n) = A356988(n).

1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 10, 11, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29

Offset: 2

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 beginning 3, 4, 5, 7, 8, 11, 13, 18, 21, 29, 34, 47, 55, ..., consists of alternating Fibonacci numbers A000045 and Lucas numbers A000032.

Links

• Peter Bala, Notes on A356993

Crossrefs

Cf. A000032, A000045, A000285, A022087, A356988, A356991 - 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( b(n - b(n - b(n - b(n)))), n = 2..100 );

Formula

a(2) = a(3) = a(4) = a(5) = 1 and then for k >= 3 there holds

a(3*F(k) + j) = F(k) for 0 <= j <= F(k-1) (local plateau)

a(L(k+1) + j) = F(k) + j for 0 <= j <= F(k-2) (ascent to plateau of height L(k-1))

a(4*F(k) + j) = L(k-1) for 0 <= j <= F(k-1) (local plateau)

a(4*F(k) + F(k-1) + j) = L(k-1) + j for 0 <= j <= F(k-3) (ascent to next plateau of height F(k+1)).