A356995 -id:A356995 - cp's OEIS frontend

A356996 a(n) = b(n) - b(b(n)) - b(n - b(n)) for n >= 3, where b(n) = A356989(n).

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 3

Peter Bala, Sep 10 2022

The sequence appears to consist of blocks of terms of the form 1, 2, 3, ..., A(k) - 1, A(k), A(k) - 1, ..., 3, 2, 1, where A(k) = A000930(k), separated by blocks of consecutive zeros.

The sequence of local peak values of the line graph of the sequence {a(n)} begins 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, ..., conjecturally A000930; the local peaks occur at abscissa values n = 8, 12, 17, 25, 37, 54, 79, 116, 170, 249, ..., conjecturally {A179070(k): k >= 7}. Cf. A356995 and A356997.

			Sequence arranged as an irregular triangle; after the first row of zeros the row lengths are conjecturally equal to A164316(k) for k >= 2.
0, 0, 0, 0, 0;
1, 0, 0, 0;
1, 0, 0, 0, 0;
1, 0, 0, 0, 0, 0, 0;
1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0;
1, 2, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
1, 2, 3, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...

Cf. A000930, A164316, A179070, A356989, A356995, A356997.

# b(n) = A356989
b := proc(n) option remember; if n = 1 then 1 else n - b(b(b(n - b(b(b(b(n-1))))))) end if; end proc:
seq(b(n) - b(b(n)) - b(n - b(n)), n = 3..300);

a(n+1) - a(n) belongs to {1, 0, -1}.

A356994 a(n) = n - b(b(b(n))), where b(n) = A356988(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 10, 10, 11, 12, 13, 14, 15, 16, 16, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 26, 26, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 42, 42, 42, 42, 42, 42, 42, 43, 44, 45

Offset: 1

Peter Bala, Sep 09 2022

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 begins 2, 4, 6, 10, 16, 26, 42, 68, 110, ..., the sequence {2*Fibonacci(k): k >= 2}
The plateau of height 2*F(k), k >= 2, has length equal to Fibonacci(k-2), starting at abscissa value n = Fibonacci(k+2) and ending at abscissa n = 3*Fibonacci(k).

Cf. A000045, A356988, A356991, A356992, A356993, A356995, A356996, A356997, A356998, A356999.

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

The sequence is determined by the initial values a(1) = 0, a(2) = 1 and the pair of formulas
1) a(n) = 2*Fibonacci(k) for n in the integer interval [Fibonacci(k+2), 3*Fibonacci(k)], k >= 2, and
2) for k >= 2, a(3*Fibonacci(k) + j) = 2*Fibonacci(k) + j for 0 <= j <= 2*Fibonacci(k-1).

cp's OEIS Frontend

A356996 a(n) = b(n) - b(b(n)) - b(n - b(n)) for n >= 3, where b(n) = A356989(n).

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