A014101 -id:A014101 - cp's OEIS frontend

A356990 a(n) = n - a^[4](n - a^[5](n-1)) with a(1) = 1, where a^[4](n) = a(a(a(a(n)))) and a^[5](n) = a(a(a(a(a(n))))).

1, 1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 19, 19, 20, 21, 22, 23, 24, 25, 26, 26, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 36, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 50, 50, 50, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 69, 69, 69, 69, 69, 69, 69, 70, 71

Offset: 1

Peter Bala, Sep 08 2022

This is the fourth sequence in a family of nested-recurrent sequences with apparently similar structure defined as follows. Given a sequence s = {s(n); n >= 1} we define the k-th iterated sequence s^[k] by putting s^[1](n) = s(n) and setting s^[k](n) = s^[k-1](u(n)) for k >= 2. For k >= 1, we define a nested-recurrent sequence {u(n): n >= 1}, dependent on k, by putting u(1) = 1 and setting u(n) = n - u^[k](n - u^[k+1](n-1)) for n >= 2. This is the case k = 4. For other cases see A006165 (k = 1), A356988 (k = 2) and A356989 (k = 3).
The sequence is slow, that is, for n >= 1, a(n+1) - a(n) is either 0 or 1.
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 5, 7, 10, 14, 19, 26, 36, 50, ..., which appears to be A003269.
The plateaus start at absiccsa values n = 6, 9, 13, 18, 24, 33, 46, 64, ..., which appears to be A014101, and terminate at abscissa values 7, 10, 14, 19, 26, 36, 50, ..., conjecturally A003269.

Cf. A003269, A006165, A014101, A356988, A356989.

a := proc(n) option remember; if n = 1 then 1 else n - a(a(a(a(n - a(a(a(a(a(n-1))))))))) end if; end proc:
seq(a(n), n = 1..100);

from functools import lru_cache
@lru_cache(maxsize=None)
def A356990(n): return 1 if n <= 1 else n-A356990(A356990(A356990(A356990(n-A356990(A356990(A356990(A356990(A356990(n-1))))))))) # Chai Wah Wu, Oct 01 2022

A136190 The 4th-order Zeckendorf array, T(n,k), read by antidiagonals.

Original entry on oeis.org

1, 2, 6, 3, 9, 8, 4, 13, 12, 11, 5, 18, 17, 16, 15, 7, 24, 23, 22, 21, 20, 10, 33, 31, 30, 29, 28, 25, 14, 46, 43, 41, 40, 39, 35, 27, 19, 64, 60, 57, 55, 54, 49, 38, 32, 26, 88, 83, 79, 76, 74, 68, 53, 45, 34, 36, 121, 114, 109, 105, 102, 93, 73, 63, 48, 37, 50, 167, 157, 150

Offset: 1

Clark Kimberling, Dec 20 2007

Rows satisfy this recurrence: T(n,k) = T(n,k-1) + T(n,k-4) for all k>=5.
Except for initial terms, (row 1) = A003269 (row 2) = A014101.
As a sequence, the array is a permutation of the natural numbers.
As an array, T is an interspersion (hence also a dispersion).

			Northwest corner:
   1  2  3  4  5  7 10  14 ...
   6  9 13 18 24 33 46  64 ...
   8 12 17 23 31 43 60  83 ...
  11 16 22 30 41 57 79 109 ...

Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Index entries for sequences that are permutations of the natural numbers

Cf. A003269 (row n=1), A134564.

Row 1 is the 4th-order Zeckendorf basis, given by initial terms b(1)=1, b(2)=2, b(3)=3, b(4)=4 and recurrence b(k) = b(k-1) + b(k-4) for k>=5. Every positive integer has a unique 4-Zeckendorf representation: n = b(i(1)) + b(i(2)) + ... + b(i(p)), where |i(h) - i(j)| >= 4. Rows of T are defined inductively: T(n,1) is the least positive integer not in an earlier row. T(n,2) is obtained from T(n,1) as follows: if T(n,1) = b(i(1)) + b(i(2)) + ... + b(i(p)), then T(n,k+1) = b(i(1+k)) + b(i(2+k)) + ... + b(i(p+k)) for k=1,2,3,... .

cp's OEIS Frontend

A356990 a(n) = n - a^[4](n - a^[5](n-1)) with a(1) = 1, where a^[4](n) = a(a(a(a(n)))) and a^[5](n) = a(a(a(a(a(n))))).

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A136190 The 4th-order Zeckendorf array, T(n,k), read by antidiagonals.

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