A356998 a(n) = b(n) - b(n - b(n)) for n >= 2, where b(n) = A356988(n).

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

Offset: 2

Author

Peter Bala, Sep 11 2022

Keywords

• nonn
• easy

Comments

The line graph of the sequence rises with slope 1 to a local peak value at heights 4, 6, 10, 16, 26, 42, ..., the sequence {2*Fibonacci(k): k >= 3}, before descending with slope -1 to a local trough at heights 3, 5, 8, 13, 21, ..., the sequence {Fibonacci(k): k >= 4}.

The local peaks of the graph occur at abscissa values n = 7, 11, 18, 29, 47, 76, ..., the sequence {Lucas(k): k >= 4}.

The trough of height F(k) starts at abscissa n = 4*F(k-1) and ends at abscissa n = F(k+2).

The sequence of trough lengths starting at abscissa n = 8 begin 0, 1, 1, 2, 3, 5, 8, 13, ..., the Fibonacci sequence A000045.

Examples

			Sequence arranged to show local peak values P and troughs T:
     0,
     1,
     2,
     2,
     3,
  P  4,
  T  3,
     4,
     5,
  P  6,
  T  5, 5,
     6,
     7,
     8,
     9,
  P  10,
     9,
  T  8, 8,
     9,
     10,
     11,
     12,
     13,
     14,
     15,
  P  16,
     15,
     14,
  T  13, 13, 13,
     14,
     15,
     16,
     17,
     18,
     19,
     20,
     21,
     22,
     23,
     24,
     25,
  P  26,
     25,
     24,
     23,
     22,
  T  21, 21, 21, 21,
     22,
     23,
     24,
     ...
		

Crossrefs

Cf. A000032, 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)), n = 1..100);

Formula

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

Let F(n) = Fibonacci(n) and L(n) + Lucas (n).

For k >= 5, a(F(k) + j) = F(k-2) + j for 0 <= j <= F(k-2) (ascent to local peak value).

For k >= 3, a(L(k)) = 2*F(k-1) (local peak values).

For k >= 4, a(L(k) + j) = 2*F(k-1) - j, for 0 <= j <= F(k-3) (descent to trough).

For k >= 2, a(4*F(k) + j) = F(k+1) for 0 <= j <= F(k-3) (local trough values).