cp's OEIS Frontend

A self-inverse permutation of the natural numbers.

a(n)=n if and only if A081242(n) is a palindrome. - Clark Kimberling, Mar 12 2003

a(n) is the position in B of the reversal of the n-th term of B, where B is the left-to-right binary enumeration sequence (A081242 with the empty word attached as first term). - Clark Kimberling, Mar 12 2003

From Antti Karttunen, Oct 28 2001: (Start)

When certain Stern-Brocot tree-related permutations are conjugated with this permutation, they induce a permutation on Z (folded to N), which is an infinite siteswap permutation (see, e.g., figure 7 in the Buhler and Graham paper, which is permutation A065174). We get:

A065260(n) = a(A057115(a(n))),

A065266(n) = a(A065264(a(n))),

A065272(n) = a(A065270(a(n))),

A065278(n) = a(A065276(a(n))),

A065284(n) = a(A065282(a(n))),

A065290(n) = a(A065288(a(n))). (End)

Every nonnegative integer has a unique representation c(1) + c(2)*2 + c(3)*2^2 + c(4)*2^3 + ..., where every c(i) is 0 or 1. Taking tuples of coefficients in lexical order (i.e., 0, 1; 01,11; 001,011,101,111; ...) yields A059893. - Clark Kimberling, Mar 15 2015

From Ed Pegg Jr, Sep 09 2015: (Start)

The reduced rationals can be ordered either as the Calkin-Wilf tree A002487(n)/A002487(n+1) or the Stern-Brocot tree A007305(n+2)/A047679(n). The present sequence gives the order of matching rationals in the other sequence.

For reference, the Calkin-Wilf tree is 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4, 1/5, 5/4, 4/7, 7/3, 3/8, 8/5, 5/7, 7/2, 2/7, 7/5, 5/8, 8/3, 3/7, 7/4, 4/5, ..., which is A002487(n)/A002487(n+1).

The Stern-Brocot tree is 1, 1/2, 2, 1/3, 2/3, 3/2, 3, 1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4, 1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, 5/4, 7/5, 8/5, 7/4, 7/3, 8/3, 7/2, ..., which is A007305(n+2)/A047679(n).

There is a great little OEIS-is-useful story here. I had code for the position of fractions in the Calkin-Wilf tree. The best I had for positions of fractions in the Stern-Brocot tree was the paper "Locating terms in the Stern-Brocot tree" by Bruce Bates, Martin Bunder, Keith Tognetti. The method was opaque to me, so I used my Calkin-Wilf code on the Stern-Brocot fractions, and got A059893. And thus the problem was solved. (End)

A self-inverse permutation of the natural numbers.

Analogous to A059893 with binary expansion replaced by minimal Fibonacci expansion.

Analogous to A343152 with maximal Fibonacci expansion replaced by minimal Fibonacci expansion.

The expansion of n equals A014417(n) with a 0 appended (see reference in link, p. 144).

Write the sequence as a (left-justified) "tetrangle" or "irregular triangle" tableau with F(t) (Fibonacci number) entries on each row, for t=1,2,3,.... Then, columns of the tableau equal rows of the Wythoff array, A035513 (see reference in link, p. 131):

1

2

3, 4

5, 7, 6

8, 11, 10, 9, 12

13, 18, 16, 15, 20, 14, 19, 17

...

Numbers k such that A007814(k) = A086784(k).

To reproduce the sequence through itself, use the following rule: if binary 1xyz is a term then so are 11xyz and 10xyz0 (except for 1 alone where 100 is not a term).

The number of terms with bit length k is equal to Fibonacci(k-1) for k > 1.

Conjecture: 2*A247648(n-1) + 1 with rewrite 1 -> 1, 01 -> 0 applied to binary expansion is the same as a(n) without trailing 0 bits in binary.

Odd terms are positive Mersenne numbers (A000225), because there is no 0 in their binary expansion. - Bernard Schott, Oct 12 2022

Permutation of the nonnegative integers.

Conjecture: A247648(n) with rewrite 1 -> 1, 01 -> 0 applied to binary expansion is the same as a(n).

Every positive integer occurs exactly once, so that, as a sequence, a(n) is a permutation of the positive integers.

Descending from the root node 1, generate tree by outer composition of L(n) = n + F(Finv(n)) and R(n) = n + F(Finv(n) + 1), respectively, according to left or right branching, where F(n) = A000045(n) are the Fibonacci numbers and Finv(n) = A130233(n) is the 'lower' Fibonacci inverse. This produces each number by maximal Fibonacci expansion (cf. example below of Method 2, entry A343152, and links).

(Level of tree): The number of terms in this expansion of n is the level of the tree on which n appears, A112310(n-1) + 1 = A200648(n+1). The number of terms in the expansion of a(n) is floor(log_2(n)) + 1 = A113473(n) = A070939(n) = A029837(n+1).

"Maximal Fibonacci expansion" maximizes the sum of coefficients over all Fibonacci numbers (of positive index), allowing both F(1) = 1 and F(2) = 1. Thus, it is just an expansion and not a representation (like "greedy" and "lazy"), as it "breaks the rule" by using two bits that correspond to elements of equal value, rather than using distinct basis elements (link). This reveals connections to the cf. sequences: Binary strings that emerge in lexicographic order from "maximal Fibonacci gaps" (example), binary trees of the positive integers, and I-D arrays "harvested" from the trees. To define the expansion uniquely, always include F(1), so that the expansion of positive integer n equals F(1) for n = 1 and F(1) prepended to the lazy Fibonacci representation of n-1 for n > 1. Hence, a(1) = 1, and for n > 1, a(n) = A095903(n-1) + 1. The "redundant" expansion arranges the positive integers in the single binary tree {T(n,k)}, rather than the two trees at A255773 and A255774 that result from representation (see link).

(Left-to-right order in tree): Each F(t)-sized block (F(t+1), ..., F(t+2) - 1) of successive positive integers ("Fibonacci cohort" t) appears in right-to-left order in the tree as reordered in A343152, where elements of each cohort appear consecutively (see link).

Descending from the root node 1, generate tree by the inner composition of A026351 and A026352, that is, one plus the sequences of lower and upper Wythoff numbers, A000201 and A001950, respectively, according to left or right branching (see example below of Method 1 and links).

Generate tree from (one plus) the number of (initial) zeros on the positive integers for the outer composition of sequences, A060143 and A060144, respectively, according to left or right branching descending from the identity (c.f example below of Method 3 and links).

The lower Wythoff numbers, A000201, appear exclusively in the 1st, 3rd, 5th, ... right clades of the tree, while the upper Wythoff numbers A001950, appear exclusively in the 2nd, 4th, 6th, ... right clades of the tree. Here, the k-th right clade comprises the nodes at positions 2^(k+1) and 2^k + 1, together with all descendants of the latter (link).

(Duality with tree A232560, and related arrays): Consider the labeled binary trees a(n) = A232560(A059893(n)) and A232560(n) = a(A059893(n)). Labels along maximal straight paths that always branch left in a(n) give rows of array A345252, while labels along maximal straight paths that always branch left in A232560 give rows of array A083047.

Sorting the labels from each successive right clade of the binary tree a(n) gives the successive columns of A083047, while sorting labels from each successive right clade of A232560 gives each successive column of A345252. This makes the trees a(n) and A232560 "blade-duals," blade being a contraction of branch-clade (see entry for A345254 and link). A200648(n)+1 gives the level of the tree on which elements of array first-columns A345252(n,1) and A083047(n,1) appear.

(Palindromes and coincidence of elements): Trees a(n) and A232560 coincide when the sequence of left and right branching is a palindrome: a(A329395(n)) = A232560(A329395(n)). As Kimberling notes (cf. A059893), this happens at fixed points of A059893(n) or, equivalently, at n for which A081242(n) is a palindrome.

The inverse permutation of a(n) as a sequence can be read from a "tetrangle" or "irregular triangle" tableau with F(t) (Fibonacci number) entries on each row t, for t = 1, 2, 3, ..., in which an entry on row t is 2*x the entry x immediately above it on row t-1, if such exists, or otherwise 2*x + 1 the entry x in the corresponding position on row t-2 (thus generating new rows as in A243571 but without sorting the numbers into increasing order, linked reference):

1,

2,

3, 4,

5, 6, 8,

7, 9, 10, 12, 16,

11, 13, 17, 14, 18, 20, 24, 32,

...

With the right-justified tableau substituted by a left-justified tableau, the same procedure yields the inverse permutation for the "minimal Fibonacci tree," A048680(A059893(n)), the "cohort-dual" tree of a(n), where "cohort" t is the F(t)-sized block of successive entries in the tableau (see entry for A345252, linked reference).

(Coincidence of elements): a(A020988(n)) = A048680(A059893(A020988(n))) = A099919(n) and a(A020989(n)) = A048680(A059893(A020989(n))) = A049651(n). Collectively, a(A061547(n)) = A048680(A059893(A061547(n))) = union(A049651(n), A099919(n)).

With two types of duality, the tree forms a quartet of binary-tree arrangements of the positive integers, together with its blade dual A232560, its cohort dual A048680(A059893), and blade dual A048680 of the latter.

Order in the tree is "memory-less": Let a(n) and a(m) label nodes at positions n and m, respectively. Let d1 and d2 be two descending paths, i.e., sequences branching left or right from a starting node. (Nodal positions for the left and right children of the node at position p are given by 2*p and 2*p + 1, resp., and d1 and d2 are compositions of these.) Then a(d1(n)) < a(d2(n)) if and only if a(d1(m)) < a(d2(m)) (linked reference).

Self-inverse permutation with swaps confined to terms of a given digit length (A134021) so within blocks n = (3^k+1)/2 .. (3^(k+1)-1)/2.

Can extend to negative n by a(-n) = -a(n).

A072998 is balanced ternary coded in decimal digits so that reversal except first digit of A072998(n) is at A072998(a(n)). Similarly its ternary equivalent A157671, and also A132141 ternary starting with 1.

These sequences all have a fixed initial digit followed by all ternary strings which is the reversed part. A007932 is such strings as decimal digits 1,2,3 but it omits the empty string so the whole reversal of A007932(n) is at A007932(a(n+1)-1).

Fixed points a(n) = n are where n in balanced ternary is a palindrome apart from its initial 1. These are the full balanced ternary palindromes with their least significant 1 removed, so all n = (A134027(m)-1)/3 for m>=2.