

A065658


The table of permutations of N, each row induced by the rotation (to the right) of the nth node in the infinite binary "decimal" fraction tree.


21



7, 25, 1, 31, 22, 1, 1, 3, 2, 1, 223, 10, 247, 2, 1, 15, 94, 4, 3, 2815, 1, 127, 6, 5, 4, 3, 2, 1, 5, 7, 28, 5, 4, 115, 2, 1, 385, 20479, 127, 6, 94, 4, 3, 2, 1, 13, 175, 8, 7, 6, 5, 4, 3, 2, 1, 1792, 46, 9, 280, 7, 234881023, 5, 4, 3, 322, 1, 61, 382, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1
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OFFSET

0,1


COMMENTS

Consider the following infinite binary tree, where the nodes are numbered in breadthfirst, lefttoright fashion from the top as in A065625 and then assigned the following rational values:
(0.1)
(0.01)(0.11)
(0.001)(0.011)(0.101)(0.111)
(0.0001)(0.0011)(0.0101)(0.0111)(0.1001)(0.1011)(0.1101)(0.1111)
i.e. the elements (1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, 1/16, 3/16, ..., of the Quasicyclic group Z+((2a+1)/(2^b)) for prime 2) listed here in their binary "decimal" fraction form. Subjecting this tree to any similar binary tree rotation as used in A065625 induces a permutation of the rationals in range ]0,1[ (i.e. including also the ones having infinite binary expansions, corresponding to infinite paths in above tree), which we then convert to permutations of N by taking the positions of the mapped values at the ]0,1[ side of the Stern Brocot Tree (A007305/A007306). See example at A065670.


LINKS

Table of n, a(n) for n=0..77.
Index entries for sequences related to Stern's sequences


MAPLE

[seq(RotateBinFracRightTable(j), j=0..119)]; RotateBinFracRightTable := n > RotateBinFracNodeRight(1+(n((trinv(n)*(trinv(n)1))/2)), (((trinv(n)1)*(((1/2)*trinv(n))+1))n)+1);
RotateBinFracNodeRight := (t, n) > frac2position_in_0_1_SB_tree(RotateBinFracNodeRight_x(t, SternBrocot0_1frac(n)));
RotateBinFracNodeRight_x := proc(t, x) local num, den; den := 2^(1+floor_log_2(t)); num := (2*(t(den/2)))+1; if((x <= (num1)/den) or (x >= (num+1)/den)) then RETURN(x); fi; if(x <= ((2*(num1))+1)/(2*den)) then RETURN((2*(x  ((num1)/den))) + ((num1)/den)); fi; if(x < (num/den)) then RETURN(x + (1/(2*den))); fi; RETURN((num/den) + ((x((num1)/den))/2)); end;
SternBrocot0_1frac := proc(n) local m; m := n + 2^floor_log_2(n); SternBrocotTreeNum(m)/SternBrocotTreeDen(m); end;
frac2position_in_0_1_SB_tree := r > RETURN(ReflectBinTreePermutation(cfrac2binexp(convert(1/r, confrac))));


CROSSREFS

The first row (rotate the top node right): A065660, 2nd row (rotate the top node's left child): A065662, 3rd row (rotate the top node's right child): A065664, 4th row: A065666, 5th row: A065668, 6th row: A065670, 7th row: A065672. For the other needed Maple procedures follow A065625, A047679, A054424 and A054429. Cf. also A065674A065676. Inverse permutations are given in A065659.
Cf. also A065934A065935.
Sequence in context: A012777 A074783 A286742 * A242322 A249437 A246792
Adjacent sequences: A065655 A065656 A065657 * A065659 A065660 A065661


KEYWORD

nonn,tabl


AUTHOR

Antti Karttunen, Nov 22 2001


STATUS

approved



