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A065658 The table of permutations of N, each row induced by the rotation (to the right) of the n-th 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Consider the following infinite binary tree, where the nodes are numbered in breadth-first, left-to-right 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 <= (num-1)/den) or (x >= (num+1)/den)) then RETURN(x); fi; if(x <= ((2*(num-1))+1)/(2*den)) then RETURN((2*(x - ((num-1)/den))) + ((num-1)/den)); fi; if(x < (num/den)) then RETURN(x + (1/(2*den))); fi; RETURN((num/den) + ((x-((num-1)/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 A065674-A065676. Inverse permutations are given in A065659.

Cf. also A065934-A065935.

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

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Last modified December 12 16:18 EST 2017. Contains 295939 sequences.