A356999 a(n) = 2*A356988(n) - n.

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

Offset: 1

Author

Peter Bala, Sep 13 2022

Keywords

• nonn
• easy

Comments

For k >= 3, the line graph of the sequence consists of a series of local peaks of height Fibonacci(k) at abscissa n = Lucas(k) separated by local valleys of height Fibonacci(k-1) at abscissa n = Fibonacci(k+2).

Examples

			The sequence arranged as a series of alternating monotone increasing and decreasing sequences:
  1;
  0;
  1, 2;
  1;
  2, 3;
  2;
  3, 4, 5;
  4, 3;
  4, 5, 6, 7, 8;
  7, 6, 5;
  6, 7, 8, 9, 10, 11, 12, 13;
  12, 11, 10, 9, 8;
  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21;
  20, 19, 18, 17, 16, 15, 14, 13;
  14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, ..., 32, 33, 34;
  33, 32, 31, ..., 23, 22, 21;
		

Crossrefs

Cf. A000032, A000045, A356988, A356991 - A356998.

Programs

Maple

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

Formula

a(n+1) - a(n) is either 1 or -1.

The sequence is completely determined by the pair of formulas

1) for k >= 2, a(F(k) + j) = F(k-3) + j for 0 <= j <= F(k-2) and

2) for k >= 1, a(L(k) + j) = F(k) - j for 0 <= j <= F(k-2),

where F(n) = A000045(n) denotes the n-th Fibonacci number (with F(-1) = 1) and L(n) = A000032(n) denotes the n-th Lucas number.