A022290 -id:A022290 - cp's OEIS frontend

A345201 Bit-reverse the odd part of the Zeckendorf representation of n: a(n) = A022290(A057889(A003714(n))).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 18, 17, 20, 21, 22, 23, 24, 30, 26, 27, 31, 29, 25, 28, 32, 33, 34, 35, 36, 37, 48, 39, 43, 49, 42, 40, 44, 50, 53, 47, 38, 41, 45, 51, 52, 46, 54, 55, 56, 57, 58, 77, 60, 69, 78, 63, 64, 70, 79, 85

Offset: 0

Rémy Sigrist, Jun 10 2021

This sequence is a self-inverse permutation of the nonnegative integers.

This sequence is similar to A343150 and to A344682.

Rémy Sigrist, Table of n, a(n) for n = 0..10946
Rémy Sigrist, PARI program for A345201
Rémy Sigrist, Alternate PARI program for A345201
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A022290, A057889, A343150, A344682.

PARI
```
See Links section.
```

a(n) < A000045(k) for any n < A000045(k).

A344682 a(0) = 0, and for any n > 0, a(n) = A022290(A059893(A003754(n+1))).

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 10, 9, 8, 11, 15, 13, 18, 12, 17, 16, 14, 19, 20, 28, 26, 23, 31, 25, 22, 30, 21, 29, 27, 24, 32, 41, 36, 49, 34, 47, 44, 39, 52, 33, 46, 43, 38, 51, 42, 37, 50, 35, 48, 45, 40, 53, 54, 75, 70, 62, 83, 68, 60, 81, 57, 78, 73, 65, 86, 67

Offset: 0

Rémy Sigrist, Jun 08 2021

This sequence is a self-inverse permutation of the nonnegative integers.

The construction of this sequence is similar to that of A343150; we start with a representation of a number n as a sum of distinct positive Fibonacci numbers, through some binary encoding, and we reverse some of the bits in a bijective way to obtain a(n).

Rémy Sigrist, Table of n, a(n) for n = 0..10944
Rémy Sigrist, PARI program for A344682
Index entries for sequences that are permutations of the natural numbers

Cf. A003754, A059893, A022290, A343150.

# after renaming a(n) to A059893(n) in A059893...
A344682 := proc(n)
    if n = 0 then
        0;
    else
        A022290(A059893(A003754(n+1))) ;
    end if;
end proc:
seq(A344682(n),n=0..70) ; # R. J. Mathar, Jul 29 2021

PARI
```
See Links section.
```

A277195 Permutation of nonnegative integers: a(n) = A022290(A277010(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 13, 9, 7, 21, 34, 10, 14, 55, 22, 89, 12, 144, 15, 35, 11, 233, 56, 23, 377, 17, 610, 90, 987, 36, 1597, 16, 57, 145, 2584, 4181, 234, 24, 25, 6765, 91, 19, 10946, 17711, 378, 18, 38, 28657, 611, 46368, 37, 988, 146, 75025, 26, 235, 1598, 58, 121393, 196418, 59, 317811, 20, 2585, 514229, 832040, 27, 379, 1346269, 93, 92

Offset: 1

Antti Karttunen, Oct 07 2016

nonn

Note the indexing: domain starts from 1, but the range includes also 0.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A277196.

Cf. A005117, A022290, A156552, A277010.

from math import isqrt
from sympy import fibonacci, mobius, primepi, factorint
def A277195(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return sum(fibonacci(primepi(p)+i) for i, p in enumerate(factorint(bisection(f), multiple=True),1)) # Chai Wah Wu, Aug 31 2024

(define (A277195 n) (A022290 (A277010 n)))

a(n) = A022290(A277010(n)) = A022290(A156552(A005117(n))).

A345101 Irregular triangle T(n, k) read by rows, n >= 0, k = 1..A000119(n); the n-th row contains the numbers m such that A022290(m) = n, in increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 16, 13, 17, 14, 18, 15, 19, 20, 21, 22, 24, 32, 23, 25, 33, 26, 34, 27, 28, 35, 36, 29, 37, 30, 38, 40, 31, 39, 41, 42, 43, 44, 48, 64, 45, 49, 65, 46, 50, 66, 47, 51, 52, 67, 68, 53, 69, 54, 56, 70, 72, 55, 57, 71, 73

Offset: 0

Rémy Sigrist, Jun 08 2021

When interpreted as a flat sequence, yields a permutation of the nonnegative integers.

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

Rémy Sigrist, Table of n, a(n) for n = 0..10922
Rémy Sigrist, PARI program for A345101
Index entries for sequences that are permutations of the natural numbers

Cf. A000119 (row lengths), A003714, A003754, A022290.

PARI
```
See Links section.
```

T(n, 1) = A003754(n+1).

T(n, A000119(n)) = A003714(n).

A361989 a(n) is the sum of the Fibonacci numbers missing from the dual Zeckendorf representation of n; a(0) = 0, and for n > 0, a(n) = A022290(A035327(A003754(n+1))).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Apr 02 2023

We consider that a Fibonacci number is missing from the dual Zeckendorf representation of a number if it does not appear in this representation and a larger Fibonacci number appears in it.
The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).
This sequence can also be seen as an irregular table T(n, k), n > 0, k = 1..A000045(n), where T(n, k) = A000045(n) - k.
a(n-1) for n>=1 is the starting position of the first occurrence of one of the longest words w in the Fibonacci word A003849 such that no length-n factor of w is repeated. The length of such words is 2n. (See links) - Gandhar Joshi, Mar 19 2024

			For n = 42:
- using F(k) = A000045(k),
- the dual Zeckendorf representation of 42 is F(8) + F(7) + F(5) + F(3) + F(2),
- the numbers F(6) and F(4) are missing,
- so a(42) = F(6) + F(4) = 8 + 3 = 11.
.
As an irregular triangle the sequence begins:
     0;
     0;
     1,  0;
     2,  1,  0;
     4,  3,  2, 1, 0;
     7,  6,  5, 4, 3, 2, 1, 0;
    12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0;
    ...

Gandhar Joshi, Walnut code and details
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A003754, A022290, A035327, A072649, A130312, A132665, A194029, A356771.

for (n = 1, 9, for (k = 1, f = fibonacci(n), print1 (f-k", ")))

a(n) = A000045(A072649(n)) - A194029(n) for n > 0.

a(n) = A130312(n) - A194029(n) for n > 0.

A364801 The number of iterations that n requires to reach a fixed point under the map x -> A022290(x).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 08 2023

a(n) is well-defined since A022290(n) = n for n <= 3 (the fixed points), and A022290(n) < n for n >= 4.

			For n = 4 the trajectory is 4 -> 3. The number of iterations is 1, thus a(4) = 1.
For n = 6 the trajectory is 6 -> 5 -> 4 -> 3. The number of iterations is 3, thus a(6) = 3.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A022290.

Similar sequences: A003434, A364800.

f[n_] := f[n] = Module[{d = IntegerDigits[n, 2], nd}, nd = Length[d]; Total[d * Fibonacci[Range[nd + 1, 2, -1]]]]; (* A022290 *)
a[n_] := -2 + Length@ FixedPointList[f, n]; Array[a, 100, 0]

f(n) = {my(b = binary(n), nb = #b); sum(i = 1, nb, b[i] * fibonacci(nb - i + 2)); } \\ A022290
a(n) = if(n < 4, 0, a(f(n)) + 1);

def A364801(n):
    if n<4: return 0
    a, b, s = 1, 2, 0
    for i in bin(n)[-1:1:-1]:
        if int(i):
            s += a
        a, b = b, a+b
    return A364801(s)+1 # Chai Wah Wu, Aug 10 2023

a(n) = a(A022290(n)) + 1, for n >= 4.

A356759 Bit-reverse the odd part of the dual Zeckendorf representation of n: a(n) = A022290(A057889(A003754(n+1))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 15, 17, 13, 16, 14, 18, 19, 20, 25, 22, 28, 30, 21, 26, 29, 23, 27, 24, 31, 32, 33, 41, 46, 36, 43, 38, 49, 51, 34, 42, 37, 47, 50, 35, 44, 48, 39, 45, 40, 52, 53, 54, 67, 59, 75, 80, 56, 70, 77, 62, 72, 64, 83, 85, 55

Offset: 0

Rémy Sigrist, Aug 26 2022

This sequence is a self-inverse permutation of the nonnegative integers, similar to A345201 and A356331.

The dual Zeckendorf (or lazy Fibonacci) representation expresses uniquely a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary, and the corresponding binary encoding, A003754(n+1), cannot have two consecutive nonleading 0's.

			For n = 49:
- the dual Zeckendorf representation of 49 is "1111010",
- reversing its odd part ("111101"), we obtain "1011110",
- so a(49) = 39.

Rémy Sigrist, Table of n, a(n) for n = 0..10945
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A003754, A022290, A057889, A104326, A345201, A356331.

PARI
```
See Links section.
```

a(a(n)) = n.

a(n) < A000045(k) iff n < A000045(k).

A360434 a(n) is the greatest number k not yet in the sequence such that A022290(n) = A022290(k).

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 8, 9, 6, 7, 10, 16, 12, 17, 18, 20, 11, 13, 14, 19, 15, 21, 32, 33, 24, 25, 34, 36, 35, 37, 40, 41, 22, 23, 26, 28, 27, 29, 38, 39, 30, 31, 42, 64, 48, 65, 66, 68, 44, 49, 50, 67, 52, 69, 72, 73, 70, 71, 74, 80, 76, 81, 82, 84, 43, 45, 46, 51

Offset: 0

Rémy Sigrist, Feb 07 2023

This sequence is a self-inverse permutation of the nonnegative integers.

			There are three numbers k such that A022290(k) = 11: 15, 19, 20,
- so a(15) = 20,
     a(19) = 19,
     a(20) = 15.

Rémy Sigrist, Table of n, a(n) for n = 0..10922
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A360415 for a similar sequence.

Cf. A000119, A003714, A003754, A022290, A345101.

PARI
```
See Links section.
```

a(A345101(n, k)) = A345101(n, A000119(n) + 1 - k).
a(A003754(n+1)) = A003714(n).
a(A003714(n)) = A003754(n+1).

A275992 Number of times n occurs in A022290.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 1, 3, 3, 3, 3, 2, 4, 4, 2, 3, 3, 3, 3, 2, 4, 4, 2, 3, 3, 3, 3, 1, 4, 4, 3, 3, 5, 4, 3, 3, 5, 5, 2, 4, 4, 4, 4, 2, 5, 5, 3, 3, 4, 4, 3, 3, 5, 5, 2, 4, 4, 4, 4, 2, 5, 5, 3, 3, 4, 5, 3, 3, 4, 4, 1, 4

Offset: 0

Max Barrentine, Aug 15 2016

nonn

A000119 counts the ways n can be represented as a sum of distinct Fibonacci numbers. A022290 maps binary ordering onto these Fibonacci representations.

A274515 is an analogous sequence applied to A002487.

Max Barrentine, Table of n, a(n) for n = 0..10922

Cf. A000119, A022290, A274515, A002487.

a(n) = A000119(A022290(n)).

A364803 Smallest number that reaches a fixed point after n iterations of the map x -> A022290(x).

Original entry on oeis.org

0, 4, 5, 6, 7, 10, 14, 23, 46, 117, 442, 3006, 47983, 2839934, 918486751, 3769839124330

Offset: 0

Amiram Eldar, Aug 08 2023

a(n) is the smallest number k such that A364801(k) = n.

Cf. A022290, A364801.

Similar sequences: A007755, A364802.

f[n_] := f[n] = Module[{d = IntegerDigits[n, 2], nd}, nd = Length[d]; Total[d * Fibonacci[Range[nd + 1, 2, -1]]]]; (* A022290 *)
iternum[n_] := -2 + Length@ FixedPointList[f, n]; (* A364801 *)
seq[kmax_] := Module[{s = {}, imax = -1, i}, Do[i = iternum[k]; If[i > imax, imax = i; AppendTo[s, k]], {k, 0, kmax}]; s]
seq[10^6]

f(n) = {my(b = binary(n), nb = #b); sum(i = 1, nb, b[i] * fibonacci(nb - i + 2)); } \\ A022290
iternum(n) = if(n < 4, 0, iternum(f(n)) + 1); \\ A364801
lista(kmax) = {my(imax = -1, i1); for(k = 0, kmax, i = iternum(k); if(i > imax, imax = i; print1(k, ", ")));}

a(15) from Martin Ehrenstein, Aug 25 2023

cp's OEIS Frontend

A345201 Bit-reverse the odd part of the Zeckendorf representation of n: a(n) = A022290(A057889(A003714(n))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A344682 a(0) = 0, and for any n > 0, a(n) = A022290(A059893(A003754(n+1))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

A277195 Permutation of nonnegative integers: a(n) = A022290(A277010(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

A345101 Irregular triangle T(n, k) read by rows, n >= 0, k = 1..A000119(n); the n-th row contains the numbers m such that A022290(m) = n, in increasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A361989 a(n) is the sum of the Fibonacci numbers missing from the dual Zeckendorf representation of n; a(0) = 0, and for n > 0, a(n) = A022290(A035327(A003754(n+1))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A364801 The number of iterations that n requires to reach a fixed point under the map x -> A022290(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A356759 Bit-reverse the odd part of the dual Zeckendorf representation of n: a(n) = A022290(A057889(A003754(n+1))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A360434 a(n) is the greatest number k not yet in the sequence such that A022290(n) = A022290(k).

Views