A344682 -id:A344682 - cp's OEIS frontend

A343150 Reverse the order of all but the most significant bits in the minimal Fibonacci expansion of n.

1, 2, 3, 4, 5, 7, 6, 8, 11, 10, 9, 12, 13, 18, 16, 15, 20, 14, 19, 17, 21, 29, 26, 24, 32, 23, 31, 28, 22, 30, 27, 25, 33, 34, 47, 42, 39, 52, 37, 50, 45, 36, 49, 44, 41, 54, 35, 48, 43, 40, 53, 38, 51, 46, 55, 76, 68, 63, 84, 60, 81, 73, 58, 79, 71, 66, 87

Offset: 1

J. Parker Shectman, Apr 07 2021

A self-inverse permutation of the natural numbers.
Analogous to A059893 with binary expansion replaced by minimal Fibonacci expansion.
Analogous to A343152 with maximal Fibonacci expansion replaced by minimal Fibonacci expansion.
The expansion of n equals A014417(n) with a 0 appended (see reference in link, p. 144).
Write the sequence as a (left-justified) "tetrangle" or "irregular triangle" tableau with F(t) (Fibonacci number) entries on each row, for t=1,2,3,.... Then, columns of the tableau equal rows of the Wythoff array, A035513 (see reference in link, p. 131):
    1
    2
    3,  4
    5,  7,  6
    8, 11, 10,  9, 12
   13, 18, 16, 15, 20, 14, 19, 17
   ...

			For an example of calculation by reversing Fibonacci binary digits, see reference in link, p. 144:
On the basis (1,1,2,3,5,8,13) n=13 is written as 0000001. Reversing all but the most significant digit gives 0000001, which evaluates to 13, so a(13)=13.
On the basis (1,1,2,3,5,8,13) n=14 is written as 0100001. Reversing all but the most significant digit gives 0000101, which evaluates to 18, so a(14)=18.
Note: The permutation can also be accomplished using the basis (1,2,3,5,8,13), by holding fixed the TWO most significant digits and reversing the remaining digits.

Cf. A014417, A035513.

In other bases: A344682 (lazy Fibonacci), A343152 (variation), A059893 (binary), A351702 (balanced ternary).

(*Produce indices of minimal Fibonacci representation (recursively)*)
MinFibInd[n_] := Module[{t = Floor[Log[GoldenRatio, Sqrt[5]*n + 1]] - 1}, Piecewise[{{{2}, n == 1}, {Append[MinFibInd[n - Fibonacci[t + 1]], t + 1], n > 1 && n - Fibonacci[t + 1] >= Fibonacci[t - 1]}, {Append[Most[MinFibInd[n - Fibonacci[t - 1]]], t + 1], n > 1 && n - Fibonacci[t + 1] < Fibonacci[t - 1]}},]];
(*Define a(n)*)
a[n_] := Module[{MFI = MinFibInd[n]}, Apply[Plus, Fibonacci[Append[Last[MFI] - Most[MFI], Last[MFI]]]]];
(*Generate DATA*)
Array[a, 67]

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

Original entry on oeis.org

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
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See Links section.
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a(n) < A000045(k) for any n < A000045(k).

A356331 Bit-reverse the odd part of the negaFibonacci representation of n: a(n) = A356327(A057889(A215024(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 10, 11, 12, 13, 14, 15, 19, 9, 18, 16, 20, 21, 51, 44, 24, 38, 26, 32, 28, 45, 46, 31, 27, 33, 34, 35, 36, 48, 25, 39, 40, 49, 53, 43, 23, 29, 30, 47, 37, 41, 50, 22, 52, 42, 54, 55, 140, 133, 58, 106, 115, 79, 62, 113, 127, 99

Offset: 0

Rémy Sigrist, Aug 04 2022

This sequence is a self-inverse permutation of the nonnegative integers similar to A343150, A344682, A345201 and A356332.

			The first terms, alongside the corresponding negaFibonacci representations, are:
  n   a(n)  nega(n)  nega(a(n))
  --  ----  -------  ----------
   0     0        0           0
   1     1        1           1
   2     2      100         100
   3     3      101         101
   4     4    10010       10010
   5     5    10000       10000
   6     6    10001       10001
   7     7    10100       10100
   8     8    10101       10101
   9    17  1001010     1010010
  10    10  1001000     1001000
  11    11  1001001     1001001
  12    12  1000010     1000010
  13    13  1000000     1000000
  14    14  1000001     1000001
  15    15  1000100     1000100
  16    19  1000101     1010001
  17     9  1010010     1001010
  18    18  1010000     1010000
  19    16  1010001     1000101
  20    20  1010100     1010100
  21    21  1010101     1010101

Rémy Sigrist, Table of n, a(n) for n = 0..6765
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, A057889, A215025, A343150, A344682, A345201, A356327, A356332.

PARI
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See Links section.
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a(a(n)) = n.

a(n) <= A000045(2*k) iff n <= A000045(2*k).

A356332 Bit-reverse the odd part of the negaFibonacci representation of -n (and negate): a(n) = -A356327(A057889(A215025(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 6, 7, 8, 9, 5, 11, 12, 31, 27, 23, 16, 17, 28, 19, 20, 21, 22, 15, 24, 30, 26, 14, 18, 29, 25, 13, 32, 33, 86, 82, 65, 71, 38, 78, 61, 57, 42, 51, 44, 45, 72, 83, 74, 62, 50, 43, 75, 53, 54, 55, 56, 41, 58, 77, 70, 40, 49, 63, 64, 36, 79, 85

Offset: 0

Rémy Sigrist, Aug 04 2022

This sequence is a self-inverse permutation of the nonnegative integers similar to A343150, A344682, A345201 and A356331.

			The first terms, alongside the corresponding negaFibonacci representations, are:
  n   a(n)  nega(-n)  nega(-a(n))
  --  ----  --------  -----------
   0     0         0            0
   1     1        10           10
   2     2      1001         1001
   3     3      1000         1000
   4     4      1010         1010
   5    10    100101       101001
   6     6    100100       100100
   7     7    100001       100001
   8     8    100000       100000
   9     9    100010       100010
  10     5    101001       100101
  11    11    101000       101000
  12    12    101010       101010
  13    31  10010101     10101001

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, A057889, A215025, A343150, A344682, A345201, A356327, A356331.

PARI
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See Links section.
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a(a(n)) = n.

a(n) < A000045(2*k+1) iff n < A000045(2*k+1).

A344962 a(n) is the position in A344953 of the reversal of the n-th word in A344953.

Original entry on oeis.org

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: 1

Clark Kimberling, Jul 10 2021

A permutation of the positive integers. Conjecture: a(n) = A344682(n) for n >= 1.

			The first twenty words w(n):  0, 1, 00, 01, 10, 000, 11, 001, 010, 100, 0000, 011, 101, 0001, 110, 0010, 0100, 1000, 00000, 111; so a(4) = 5.

Cf. A341258, A342910.

Mathematica
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(See A341258.)
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cp's OEIS Frontend

A343150 Reverse the order of all but the most significant bits in the minimal Fibonacci expansion of n.

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

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A356331 Bit-reverse the odd part of the negaFibonacci representation of n: a(n) = A356327(A057889(A215024(n))).

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A356332 Bit-reverse the odd part of the negaFibonacci representation of -n (and negate): a(n) = -A356327(A057889(A215025(n))).

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A344962 a(n) is the position in A344953 of the reversal of the n-th word in A344953.

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