A357563 a(n) = b(n) - 2*b(b(b(n))) for n >= 3, where b(n) = A356988(n).

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

Offset: 3

Author

Peter Bala, Oct 14 2022

Keywords

• nonn
• easy

Comments

a(n+1) - a(n) is equal to 0, 1 or -1.

The sequence vanishes at abscissa values n = 3, 6, 9, 15, 24, 39, ..., 3*Fibonacci(k), ....

For k >= 2, the line graph of the sequence, starting from the zero value at abscissa n = 3*Fibonacci(k), ascends with slope 1 to a local plateau at height Fibonacci(k-1) at abscissa value n = Lucas(k+1). The plateau has length Fibonacci(k-1). From the end of the plateau, at abscissa value n = Fibonacci(k+3), the graph of the sequence descends with slope -1 to the next zero at abscissa n = 3*Fibonacci(k+1).

Links

• Peter Bala, Notes on A357563

Crossrefs

Cf. A000032, A000045, A356988, A357562.

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( b(n) - 2*b(b(b(n))), n = 3..100);

Formula

For k >= 2 there holds

a(3*Fibonacci(k) + j) = j for 0 <= j <= Fibonacci(k-1) (rise from 0 to plateau)

a(Lucas(k+1) + j) = Fibonacci(k-1) for 0 <= j <= Fibonacci(k-1) (plateau)

a(Fibonacci(k+3) + j) = Fibonacci(k-1) - j for 0 <= j <= Fibonacci(k-1) (descent back to 0).