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

1,2

Comments

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

...

Links

Table of n, a(n) for n=1..67.

J. Parker Shectman, A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration

Index entries for sequences that are permutations of the natural numbers

Example

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.

Mathematica

(*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]

Crossrefs

Cf. A014417, A035513.

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

Sequence in context: A273751 A056017 A091995 * A338698 A361946 A066937

Adjacent sequences: A343147 A343148 A343149 * A343151 A343152 A343153

Keyword

nonn,base,easy

Author

J. Parker Shectman, Apr 07 2021

Status

approved