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A343152
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Reverse the order of all but the most significant bits in the maximal Fibonacci expansion of n.
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6
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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
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OFFSET
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1,2
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COMMENTS
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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
...
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LINKS
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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
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EXAMPLE
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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.
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MATHEMATICA
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(*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]
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CROSSREFS
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Cf. A059893, A083047, A104326, A247647, A343150.
Sequence in context: A330081 A072759 A257683 * A267105 A080542 A330090
Adjacent sequences: A343149 A343150 A343151 * A343153 A343154 A343155
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KEYWORD
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nonn,easy,base
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AUTHOR
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J. Parker Shectman, Apr 07 2021
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STATUS
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approved
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