A343152 Reverse the order of all but the most significant bits in the maximal Fibonacci expansion of n.

1, 2, 3, 4, 6, 5, 7, 8, 11, 10, 9, 12, 16, 14, 19, 13, 18, 17, 15, 20, 21, 29, 27, 24, 32, 26, 23, 31, 22, 30, 28, 25, 33, 42, 37, 50, 35, 48, 45, 40, 53, 34, 47, 44, 39, 52, 43, 38, 51, 36, 49, 46, 41, 54, 55, 76, 71, 63, 84, 69, 61, 82, 58, 79, 74, 66, 87

Offset

1,2

Comments

A self-inverse permutation of the natural numbers.

Analogous to A059893 with binary expansion replaced by maximal Fibonacci expansion.

Analogous to A343150 with minimal Fibonacci expansion replaced by maximal Fibonacci expansion.

For n=1, the expansion equals 1. For n>=2, the expansion equals A104326(n-1) with a 1 appended. The 1 corresponds to a digit (always equal to 1) for F(1)=1, in addition to the digit for F(2)=1. (This expansion is NOT a representation, see reference in link, pp. 106 and 137.)

Write the sequence as a (right-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 array A083047 (see reference in link, p. 131):

1

2

3, 4

6, 5, 7

8, 11, 10, 9, 12

16, 14, 19, 13, 18, 17, 15, 20

...

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) n=13 is written as 110101, Reversing all but the most AND least significant digits gives 101011, which evaluates to 16, so a(13)=16.
On the basis (1,1,2,3,5,8) n=14 is written as 101101, Reversing all but the most AND least significant digits gives 101101, which evaluates to 14, so a(14)=14.

Mathematica

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

Crossrefs

Cf. A059893, A083047, A104326, A247647, A343150.

Sequence in context: A072759 A361482 A257683 * A267105 A080542 A330090

Adjacent sequences: A343149 A343150 A343151 * A343153 A343154 A343155

Keyword

nonn,easy,base

Author

J. Parker Shectman, Apr 07 2021

Status

approved