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

Author

J. Parker Shectman, Apr 07 2021

Keywords

• nonn
• base
• easy

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

...

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A014417, A035513.

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

Programs

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]