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
Examples
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.
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.
Crossrefs
Programs
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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}]