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A339132 Milk shuffle of the binary representation of n.

%I A339132 #35 Jan 04 2021 08:44:21
%S A339132 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,
%T A339132 26,27,30,31,2,3,6,7,18,19,22,23,34,35,38,39,50,51,54,55,10,11,14,15,
%U A339132 26,27,30,31,42,43,46,47,58,59,62,63,2,3,6,7,18,19,22,23
%N A339132 Milk shuffle of the binary representation of n.
%H A339132 Sander G. Huisman, <a href="/A339132/b339132.txt">Table of n, a(n) for n = 0..5000</a> [a(0)=0 inserted by _Georg Fischer_, Jan 04 2021]
%H A339132 Roger Antonsen, <a href="https://archive.bridgesmathart.org/2018/bridges2018-451.html">Card Shuffling Visualizations</a>, Bridges Conference Proceedings, 2018.
%e A339132 For n = 19 we take the binary representation without leading zeros: 10011.
%e A339132 We now shuffle the binary digits around according to A209279, which can be interpreted as a so-called milk shuffle.
%e A339132 For five digits the n-th digits gets moved around as follows: 1,2,3,4,5 => 3,2,4,1,5.
%e A339132 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.
%e A339132 We now apply this reshuffling to our binary digits of 19: 00111.
%e A339132 This is now reinterpreted into a decimal number: 7.
%t A339132 milk[list_]:=Table[list[[{i,-i}]],{i,Length[list]/2}]//milkPost[#,list]&//Reverse//Flatten
%t A339132 milkPost[x_,list_]:=x/;EvenQ[Length[list]]
%t A339132 milkPost[x_,list_]:=Join[x,{list[[(Length[list]+1)/2]]}]
%t A339132 Table[FromDigits[milk@IntegerDigits[i,2],2],{i,0,500}]
%t A339132 (*OR*)
%t A339132 Table[FromDigits[ResourceFunction["Shuffle"][IntegerDigits[i,2],"Milk"],2], {i,0,500}]
%Y A339132 Cf. A330090 (shuffle bits low to high).
%Y A339132 Cf. A209279 (1-based shuffle), A332104 (0-based shuffle).
%K A339132 base,easy,look,nonn
%O A339132 0,3
%A A339132 _Sander G. Huisman_, Nov 24 2020