A330090 -id:A330090 - cp's OEIS frontend

A330081 If the binary expansion of n is (b(1), ..., b(w)), then the binary expansion of a(n) is (b(1), b(3), b(5), ..., b(6), b(4), b(2)).

0, 1, 2, 3, 4, 6, 5, 7, 8, 10, 12, 14, 9, 11, 13, 15, 16, 20, 18, 22, 24, 28, 26, 30, 17, 21, 19, 23, 25, 29, 27, 31, 32, 36, 40, 44, 34, 38, 42, 46, 48, 52, 56, 60, 50, 54, 58, 62, 33, 37, 41, 45, 35, 39, 43, 47, 49, 53, 57, 61, 51, 55, 59, 63, 64, 72, 68, 76

Offset: 0

Rémy Sigrist, Dec 01 2019

This sequence is a permutation of the nonnegative integers that preserves the binary length as well as the Hamming weight. See A330090 for the inverse.

			For n = 1234:
- the binary expansion of 1234 is "10011010010",
- odd-indexed bits are "101100",
- even-indexed bits are "01001", and in reverse order "10010",
- hence the binary expansion of a(1234) is "10110010010",
- so a(1234) = 1426.

See A329303 for a similar sequence.

Cf. A003558, A194959, A330090 (inverse).

shuffle(v) = { my (w=vector(#v), o=0, e=#v+1); for (k=1, #v, w[if (k%2, o++, e--)]=v[k]); w }
a(n) = fromdigits(shuffle(binary(n)), 2)

If n has w binary digits, then a^A003558(w-1)(n) = n (where a^k denotes the k-th iterate of the sequence).

A339132 Milk shuffle of the binary representation of n.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 7, 2, 3, 6, 7, 10, 11, 14, 15, 2, 3, 6, 7, 18, 19, 22, 23, 10, 11, 14, 15, 26, 27, 30, 31, 2, 3, 6, 7, 18, 19, 22, 23, 34, 35, 38, 39, 50, 51, 54, 55, 10, 11, 14, 15, 26, 27, 30, 31, 42, 43, 46, 47, 58, 59, 62, 63, 2, 3, 6, 7, 18, 19, 22, 23

Offset: 0

Sander G. Huisman, Nov 24 2020

			For n = 19 we take the binary representation without leading zeros: 10011.
We now shuffle the binary digits around according to A209279, which can be interpreted as a so-called milk shuffle.
For five digits the n-th digits gets moved around as follows: 1,2,3,4,5 => 3,2,4,1,5.
This reshuffling can be thought of taking the middle number, and then alternatingly taking digits from the left and then the right until all digits are taken.
We now apply this reshuffling to our binary digits of 19: 00111.
This is now reinterpreted into a decimal number: 7.

Sander G. Huisman, Table of n, a(n) for n = 0..5000 [a(0)=0 inserted by _Georg Fischer_, Jan 04 2021]
Roger Antonsen, Card Shuffling Visualizations, Bridges Conference Proceedings, 2018.

Cf. A330090 (shuffle bits low to high).

Cf. A209279 (1-based shuffle), A332104 (0-based shuffle).

milk[list_]:=Table[list[[{i,-i}]],{i,Length[list]/2}]//milkPost[#,list]&//Reverse//Flatten
milkPost[x_,list_]:=x/;EvenQ[Length[list]]
milkPost[x_,list_]:=Join[x,{list[[(Length[list]+1)/2]]}]
Table[FromDigits[milk@IntegerDigits[i,2],2],{i,0,500}]
(*OR*)
Table[FromDigits[ResourceFunction["Shuffle"][IntegerDigits[i,2],"Milk"],2], {i,0,500}]

cp's OEIS Frontend

A330081 If the binary expansion of n is (b(1), ..., b(w)), then the binary expansion of a(n) is (b(1), b(3), b(5), ..., b(6), b(4), b(2)).

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