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

%I A343150 #56 Feb 25 2022 07:06:09
%S A343150 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,
%T A343150 31,28,22,30,27,25,33,34,47,42,39,52,37,50,45,36,49,44,41,54,35,48,43,
%U A343150 40,53,38,51,46,55,76,68,63,84,60,81,73,58,79,71,66,87
%N A343150 Reverse the order of all but the most significant bits in the minimal Fibonacci expansion of n.
%C A343150 A self-inverse permutation of the natural numbers.
%C A343150 Analogous to A059893 with binary expansion replaced by minimal Fibonacci expansion.
%C A343150 Analogous to A343152 with maximal Fibonacci expansion replaced by minimal Fibonacci expansion.
%C A343150 The expansion of n equals A014417(n) with a 0 appended (see reference in link, p. 144).
%C A343150 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):
%C A343150     1
%C A343150     2
%C A343150     3,  4
%C A343150     5,  7,  6
%C A343150     8, 11, 10,  9, 12
%C A343150    13, 18, 16, 15, 20, 14, 19, 17
%C A343150    ...
%H A343150 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 A343150 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A343150 For an example of calculation by reversing Fibonacci binary digits, see reference in link, p. 144:
%e A343150 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.
%e A343150 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.
%e A343150 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.
%t A343150 (*Produce indices of minimal Fibonacci representation (recursively)*)
%t A343150 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]}},]];
%t A343150 (*Define a(n)*)
%t A343150 a[n_] := Module[{MFI = MinFibInd[n]}, Apply[Plus, Fibonacci[Append[Last[MFI] - Most[MFI], Last[MFI]]]]];
%t A343150 (*Generate DATA*)
%t A343150 Array[a, 67]
%Y A343150 Cf. A014417, A035513.
%Y A343150 In other bases: A344682 (lazy Fibonacci), A343152 (variation), A059893 (binary), A351702 (balanced ternary).
%K A343150 nonn,base,easy
%O A343150 1,2
%A A343150 _J. Parker Shectman_, Apr 07 2021