A257676 Permutation of nonnegative integers obtained by traversing the tendrils (finite side-trees) of the binary beanstalk in depth-first order, with also each number in the infinite trunk visited, but only after its sister branch has been traversed first.

0, 1, 2, 3, 5, 4, 6, 7, 9, 8, 10, 12, 13, 11, 14, 15, 17, 16, 18, 20, 21, 19, 22, 24, 25, 28, 29, 23, 27, 26, 30, 31, 33, 32, 34, 36, 37, 35, 38, 40, 41, 44, 45, 39, 43, 42, 47, 50, 54, 58, 59, 55, 51, 46, 48, 49, 52, 53, 56, 60, 61, 57, 62, 63, 65, 64, 66, 68, 69, 67, 70, 72, 73, 76, 77, 71, 75, 74, 79, 82, 86, 90, 91, 87, 83, 78, 80, 81

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..16384

Antti Karttunen, Same Scheme-function as in Program section, but indented and commented properly

Paul Tek, Illustration of how natural numbers in range 0 .. 133 are organized as a binary tree in the binary beanstalk

Index entries for sequences that are permutations of the natural numbers

Formula

a(0) = 0; a(1) = 1;

otherwise set prev = a(n-1);

if A213719(prev) = 1 [prev is one of the terms in A179016]

then if A213719(A213723(prev)) = 0, a(n) = A213723(prev),

else a(n) = A213724(prev);

else if(A213723(prev) > 0), a(n) = A213723(prev),

else if(A213724(prev) > 0), a(n) = A213724(prev),

otherwise,

a(n) = {the first unvisited node of binary beanstalk tree found when we backtrack out of a finite branch just traversed in depth-first order}.

Other identities and observations:

If a(n-1) is an even term of A055938 then a(n) = a(n-1)+1.

Example

Please look at Paul Tek's illustration: We start at root, 0, go up to 1, visit its left child 2 (which is a leaf), before proceeding the infinite trunk (A179016) to 3, then visit first the leaf 5 at the right hand side, before proceeding the infinite trunk to 4, then visit the leaf 6 at the left hand side, before proceeding the infinite trunk right to 7, from which we first visit the leaf 9 at the right hand side, before proceeding the infinite trunk to 8 at the left hand side. Thus we have ten initial terms of the sequence: 0, 1, 2, 3, 5, 4, 6, 7, 9, 8, ...
From 8 we proceed first to the left 10, because it is not a part of the infinite trunk, and we traverse a finite side-tree ("tendril") of three nodes in order 10, 12, 13, only after which we proceed the infinite trunk to the right, to 11, thus we have the next four terms of the sequence 10, 12, 13, 11.

Prog

(Scheme, with defineperm1-macro from Antti Karttunen's IntSeq-library)
(defineperm1 (A257676 n) (if (<= n 1) n (let ((prev (A257676 (- n 1)))) (cond ((= 1 (A213719 prev)) (if (zero? (A213719 (A213723 prev))) (A213723 prev) (A213724 prev))) ((not (zero? (A213723 prev))) (A213723 prev)) ((not (zero? (A213724 prev))) (A213724 prev)) (else (let loop ((prev prev) (back (A011371 prev))) (cond ((= 1 (A213719 back)) (if (zero? (A213719 (A213723 back))) (A213724 back) (A213723 back))) ((and (even? prev) (not (zero? (A213724 back)))) (A213724 back)) (else (loop back (A011371 back))))))))))
;; Please see the attached text file for the same code better indented and documented.

Crossrefs

Inverse: A257677.

Fixed points: A257678.

Cf. A011371, A055938, A179016, A213719, A213723, A213724, A213725, A213727, A213730.

Cf. also A218252.

Sequence in context: A130981 A074145 A299237 * A257677 A105363 A279337

Adjacent sequences: A257673 A257674 A257675 * A257677 A257678 A257679

Keyword

nonn

Author

Antti Karttunen, May 04 2015

Status

approved