A356997 a(n) = b(n) - b(n - b(n - b(n))) for n >= 2, where b(n) = A356988(n).

0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 3, 3, 3, 2, 2, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 4, 3, 3, 3, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 5, 5, 5, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 12, 11, 10, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11

Offset: 2

Author

Peter Bala, Sep 11 2022

Keywords

• nonn
• easy

Comments

The line graph of the sequence consists of a series of local plateaus and local troughs joined at each end by lines of slope 1 and slope -1. More precisely, for k >= 3 the graph of the sequence consists of

a) local plateaus: on the integer interval [2*F(k), 2*F(k) + 2*F(k-3)] the sequence has the constant value F(k-2), where F(n) denotes the n-th Fibonacci number

b) descent to a trough: on the integer interval [2*F(k) + 2*F(k-3), F(k+2)] the line graph of the sequence has slope -1

c) local troughs: on the integer interval [F(k+2), F(k+2) + F(k-3)] the sequence has the constant value F(k-3)

d) ascent to a plateau: on the integer interval [F(k+2) + F(k-3), 2*F(k+1)] the line graph of the sequence has slope 1.

Examples

			The sequence is arranged to show the local plateaus (P) and the local troughs (T):
    0,
    1,
    1,
T   0,
P   1, 1, 1
    1,
P   2, 2, 2,
T   1,1,
    2,
P   3, 3, 3, 3, 3,
T   2, 2, 2,
    3,
    4,
P   5, 5, 5, 5, 5, 5, 5,
    4,
T   3, 3, 3, 3,
    4,
    5,
    6,
    7,
P   8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
    7,
    6,
T   5, 5, 5, 5, 5, 5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
P   13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
    12,
    11,
    10,
    9,
T   8, 8, 8, 8, 8, 8, 8, 8, 8,
    9,
    10,
    11,
    ...
		

Crossrefs

Cf. A000045, 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(n+1) - a(n) = 1, 0 or -1.

Let F(n) = A000045(n) with F(-1) = 1 and let L(n) = A000032(n).

For k >= 5, a(F(k) + j) = F(k-5) for 0 <= j <= F(k-5) (troughs).

For k >= 4, a(2*F(k) + j) = F(k-2) for 0 <= j <= 2*F(k-3) (plateaus).