cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

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

Original entry on oeis.org

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

Views

Author

Peter Bala, Sep 13 2022

Keywords

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.

A356993 a(n) = b(n - b(n - b(n - b(n)))) for n >= 2, where b(n) = A356988(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 10, 11, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29
Offset: 2

Views

Author

Peter Bala, Sep 09 2022

Keywords

Comments

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 beginning 3, 4, 5, 7, 8, 11, 13, 18, 21, 29, 34, 47, 55, ..., consists of alternating Fibonacci numbers A000045 and Lucas numbers A000032.

Links

Crossrefs

Cf. A000032, A000045, A000285, A022087, 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(2) = a(3) = a(4) = a(5) = 1 and then for k >= 3 there holds
a(3*F(k) + j) = F(k) for 0 <= j <= F(k-1) (local plateau)
a(L(k+1) + j) = F(k) + j for 0 <= j <= F(k-2) (ascent to plateau of height L(k-1))
a(4*F(k) + j) = L(k-1) for 0 <= j <= F(k-1) (local plateau)
a(4*F(k) + F(k-1) + j) = L(k-1) + j for 0 <= j <= F(k-3) (ascent to next plateau of height F(k+1)).

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

Original entry on oeis.org

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

Views

Author

Peter Bala, Sep 11 2022

Keywords

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).

A357562 a(n) = n - 2*b(b(n)) for n >= 2, where b(n) = A356988(n).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12
Offset: 2

Views

Author

Peter Bala, Oct 14 2022

Keywords

Comments

a(n+1) - a(n) is equal to 1 or -1.
The sequence vanishes at abscissa values n = 2, 4, 6, 10, 16, 26, ..., 2*Fibonacci(k), .... For k >= 2, the line graph of the sequence, starting from the zero value at abscissa n = 2*Fibonacci(k), ascends with slope 1 to a local peak at height Fibonacci(k-1) at abscissa value n = Fibonacci(k+2) before descending with slope -1 to the next zero at abscissa n = 2*Fibonacci(k+1).
a(n) = the distance to the nearest number of the form 2*Fibonacci(k). Cf. A053646.

Links

Crossrefs

Cf. A000045, A053646, A356988, A357563, A357564.

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

Formula

For k >= 2 there holds
a(2*Fibonacci(k) + j ) = j for 0 <= j <= Fibonacci(k-1) and
a(Fibonacci(k+2) + j) = Fibonacci(k-1) - j for 0 <= j <= Fibonacci(k-1).

A356992 Then a(n) = n - b(n - b(n - b(n - b(n - b(n - b(n)))))) for n >= 2, where b(n) = A356988(n).

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 5, 6, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 48
Offset: 2

Views

Author

Peter Bala, Sep 08 2022

Keywords

Comments

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 4, 7, 11, 18, 29, 47, ..., conjecturally the Lucas sequence {A000032(k): k >= 3}.
The plateaus start at abscissa values n = 5, 10, 16, 26, 42, 68, .... Apart from the first term 5, this appears to be the sequence {2*Fibonacci(k): k >= 5}.
The plateaus end at abscissa values n = 7, 12, 19, 31, 50, 81, ..., conjecturally the sequence {A013655(k): k >= 3}.
The sequence of plateau lengths begins 2, 2, 3, 5, 8, 13, .... Apart from the first term 2, this appears to be the sequence {Fibonacci(k): k >= 3}.
The slow sequences {a(a(n)): n >= 3} and {a(a(a(n))): n >= 4} appear to have similar properties to the present sequence. The slow sequence {n - a(n): n >= 2} appears to have plateaus at heights given by the Fibonacci sequence. See the Example section.

Examples

			Related sequences:
1) {n - a(n): n >= 2}
  1, 1, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 35, 36, 37, ...
The line graph of the sequence has plateaus at heights 3, 5, 8, 13, 21, 34, ..., conjecturally the Fibonacci numbers A000045.
2) {a(a(n)): n >= 3}
  1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 47, 47, 47, 47, ...
The line graph of the sequence has plateaus at heights 3, 4, 7, 11, 28, 29, ..., conjecturally the Lucas numbers A000045.
3) {a(a(a(n))): n >= 4}
  1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29, ...
The line graph of the sequence has plateaus at heights (2), 3, 4, 7, 11, 28, 29, ..., conjecturally the Lucas numbers A000045.

Crossrefs

Cf. A000032, A000045, A013655, A356988, A356991 - A356999.

Programs

  • Maple
    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(n - b(n - b(n - b(n - b(n - b(n)))))), n = 2..100);

A356995 a(n) = b(n) - b(b(n)) - b(n - b(n)) for n >= 3, where b(n) = A356988(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 1, 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, 5, 4, 3, 2, 1, 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, 7, 6, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0
Offset: 3

Views

Author

Peter Bala, Sep 09 2022

Keywords

Comments

Starting at n = 7, the sequence consists of successive blocks of integers of the form 1, 2, 3, ..., F(k) - 1, F(k), F(k) - 1, ..., 3, 2, 1, where F(k), k >= 1, denotes the k-th Fibonacci number, followed by a string of zeros conjecturally of length 1 + 2*F(k+1).
The sequence has local peak values at abscissa values n = 7, 11, 18, ..., L(k), ..., where L(k) = A000032(k), the k-th Lucas number. The zero strings begin at abscissa values n = 8, 12, 20, 32, 52, ..., equal to the sequence {L(k) + F(k-3) : k >= 4} = {4*F(k-1): k >= 4}.

Examples

			Sequence {a(n)} arranged as a sequence of strings of length 2*Fibonacci(k), k >= 1
  0, 0;
  0, 0;
  1, 0, 0, 0;
  1, 0, 0, 0, 0, 0;
  1, 2, 1, 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, 5, 4, 3, 2, 1, 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, 7, 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, 0, 0, 0, 0;
  ...

Crossrefs

Cf. A000032, A000045, A356988, A356991 - A356999.

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) - b(b(n)) - b(n - b(n)), n = 3..250);

Formula

a(n+1) - a(n) is in {1, 0, -1}.
For k >= 3, a(L(k) + j) = F(k-3) - j and a(L(k) - j) = F(k-3) - j for 0 <= j <= F(k-3), where F(k) = A000045(k), the k-th Fibonacci number and L(k) = A000032(k), the k-th Lucas number.

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

Original entry on oeis.org

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

Views

Author

Peter Bala, Sep 11 2022

Keywords

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).

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

Original entry on oeis.org

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

Views

Author

Peter Bala, Oct 14 2022

Keywords

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

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).

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

Views

Author

Peter Bala, Sep 09 2022

Keywords

Comments

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).

Crossrefs

Cf. A000045, A356988, A356991, A356992, A356993, A356995, A356996, A356997, A356998, 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( n - b(b(b(n))), n = 1..100);

Formula

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).

A356991 a(n) = b(n) + b(n - b(n)) for n >= 2, where b(n) = A356998(n).

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 47, 47, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 76, 76, 76, 76, 76, 77, 78, 79, 80
Offset: 2

Views

Author

Peter Bala, Sep 08 2022

Keywords

Comments

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) is unchanged with increasing values of the abscissa n) joined by lines of slope 1.
The sequence of plateau heights begins 4, 7, 11, 18, 29, 47, 76, 123, 199, ..., the Lucas sequence {A000032(k): k >= 3}. The plateaus start at absiccsa values n = 4, 8, 12, 20, 32, 52, 84, 136, ..., the sequence {A022087(k): k >= 2}, and end at abscissa values n = 5, 8, 13, 21, 34, 55, 89, ..., the Fibonacci sequence {A000045(k): k >= 5}.
Compare with A356992 and A356993.
Other sequences defined in terms of b(n) = A356998(n) that are similarly related to the Lucas numbers include {n - b(b(b(2*n - b(n)))): n >= 1} beginning 0, 1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 17, 18, 19, ... and {2*n - b(2*n - b(2*n - b(n))) : n >= 1} beginning 1, 3, 4, 5, 7, 7, 9, 11, 11, 12, 14, 16, 18, 18, 18, 19, 21, 23, 25, 27, 29, 29, 29, 29, 29, 31, .... Neither sequence is slow.

Crossrefs

Cf. A000032, A000045, A022087, A356988, A356992 - A356999.

Programs

  • Maple
    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 = 2..100);

Formula

The sequence is completely determined by the initial values a(2) = 2, a(3) = 3 and the pair of formulas:
1) for k >= 3, a(4*F(k-1) + j) = L(k) for 0 <= j <= F(k-4), where F(-1) = 1 and
2) for k >= 3, a(F(k+2) + j) = L(k) + j for 0 <= j <= L(k-1).
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