%I

%S 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,

%T 23,31,22,30,28,25,33,42,37,50,35,48,45,40,53,34,47,44,39,52,43,38,51,

%U 36,49,46,41,54,55,76,71,63,84,69,61,82,58,79,74,66,87

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

%C A self-inverse permutation of the natural numbers.

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

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

%C 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.)

%C 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):

%C 1

%C 2

%C 3, 4

%C 6, 5, 7

%C 8, 11, 10, 9, 12

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

%C ...

%H J. Parker Shectman, <a href="http://www.ootlinc.com/Fibonacci_Quilt_2_of_3_Cohorts_and_Numeration.pdf">A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e For an example of calculation by reversing Fibonacci binary digits, see reference in link, p. 144:

%e 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.

%e 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.

%t (*Produce indices of maximal Fibonacci expansion (recursively)*)

%t MaxFibInd[n_] := Module[{t = Floor[Log[GoldenRatio, Sqrt[5]*n + 1]] - 1}, Piecewise[{{{1}, n == 1}, {Append[MaxFibInd[n - Fibonacci[t]], t], n > 1}},]];

%t (*Define a(n)*)

%t a[n_] := Module[{MFI = MaxFibInd[n]}, Apply[Plus, Fibonacci[Last[MFI] - MFI + 1]]];

%t (*Generate DATA*)

%t Array[a, 67]

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

%K nonn,easy,base

%O 1,2

%A _J. Parker Shectman_, Apr 07 2021