A339132 Milk shuffle of the binary representation of n.

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

0,3

Links

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.

Example

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.

Mathematica

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}]

Crossrefs

Cf. A330090 (shuffle bits low to high).

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

Sequence in context: A079025 A165930 A300500 * A064895 A120877 A326304

Adjacent sequences: A339129 A339130 A339131 * A339133 A339134 A339135

Keyword

base,easy,look,nonn

Author

Sander G. Huisman, Nov 24 2020

Status

approved