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The inverse permutation is A284460.

The inverse permutation is A284459.

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.

Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - Labos Elemer, May 07 2002

Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940. - Reinhard Zumkeller, Feb 25 2003

Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - Benoit Cloitre, Mar 12 2003

The sequence of first differences is this sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005

Subsequence of A122132. - Reinhard Zumkeller, Aug 21 2006

Apart from the first term, a subsequence of A124509. - Reinhard Zumkeller, Nov 04 2006

Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007

Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005

For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} -> {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n). - Milan Janjic, May 10 2007

a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953). - Jaroslav Krizek, Aug 17 2009

Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010

Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012

a(n-1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212. - Jon Perry, Oct 10 2012

If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x-4. This equation also has solutions 84, 3348, 1450092, ... which are not of the form 3*2^n. - Farideh Firoozbakht, Nov 30 2013

a(n) is the upper bound for the "X-ray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu). - L. Edson Jeffery, Jan 11 2014

If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n-2) many open sets. See Brown and Stephen references. - Ross La Haye, Jan 19 2014

Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)

Fix a dimension n. For a real-valued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all k-point subsets S in E. Moreover, the best possible k is 3 * 2^(n-1).

The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.

For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)

Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015

The average of consecutive powers of 2 beginning with 2^1. - Melvin Peralta and Miriam Ong Ante, May 14 2016

For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207). - Ivan N. Ianakiev, Dec 09 2016

Also, for n >= 2, the number of length-n strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit) - Jeffrey Shallit, Dec 02 2019

Also, a(n) is the minimum link-length of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)-dimensional hypercube. - Marco Ripà, Aug 22 2022

The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - Antti Karttunen, Sep 06 2023

The finite subsequence a(3), a(4), a(5), a(6) = 24, 48, 96, 192 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A000244 (see comment there). - Felix Huber, Feb 15 2024

A level 1 Sierpiński triangle is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles. For n>2, a(n-3) is the radius of the level n Sierpiński triangle graph. - Allan Bickle, Aug 03 2024

a(n) gives the position of the inverse of the n-th term in the full Stern-Brocot tree: A007305(a(n)+2) = A047679(n) and A047679(a(n)) = A007305(n+2). - Reinhard Zumkeller, Dec 22 2008

From Gary W. Adamson, Jun 21 2012: (Start)

The mapping and conversion rules are as follows:

By rows, we have ...

1;

3, 2;

7, 6, 5, 4;

15, 14, 13, 12, 11, 10, 9, 8;

... onto which we are to map one-half of the Stern-Brocot infinite Farey Tree:

1/2

1/3, 2/3

1/4, 2/5, 3/5, 3/4

1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5

...

The conversion rules are: Convert the decimal to binary, adding a duplicate of the rightmost binary term to its right. For example, 10 = 1010, which becomes 10100. Then, from the left, record the number of runs = [1,1,1,2], the continued fraction representation of 5/8. Check: 10 decimal corresponds to 5/8 as shown in the overlaid mapping. Take decimal 9 = 1001 which becomes 10011, with a continued fraction representation of [1,2,2] = 5/7. Check: 9 decimal corresponds to 5/7 in the Farey Tree map. (End)

From Indranil Ghosh, Jan 19 2017: (Start)

a(n) is the value generated when n is converted into its Elias gamma code, the 1's and 0's are interchanged and the resultant is converted back to its decimal value for all values of n > 1. For n = 1, A054429(n) = 1 but after converting 1 to Elias gamma code, interchanging the 1's and 0's and converting it back to decimal, the result produced is 0.

For example, let n = 10. The Elias gamma code for 10 is '1110010'. After interchanging the 1's and 0's it becomes "0001101" and 1101_2 = 13_10. So a(10) = 13. (End)

From Yosu Yurramendi, Mar 09 2017 (similar to Zumkeller's comment): (Start)

A002487(a(n)) = A002487(n+1), A002487(a(n)+1) = A002487(n), n > 0.

A162909(a(n)) = A162910(n), A162910(a(n)) = A162909(n), n > 0.

A162911(a(n)) = A162912(n), A162912(a(n)) = A162911(n), n > 0.

A071766(a(n)) = A245326(n), A245326(a(n)) = A071766(n), n > 0.

A229742(a(n)) = A245325(n), A245325(a(n)) = A229742(n), n > 0.

A020651(a(n)) = A245327(n), A245327(a(n)) = A020651(n), n > 0.

A020650(a(n)) = A245328(n), A245328(a(n)) = A020650(n), n > 0. (End)

From Yosu Yurramendi, Mar 29 2017: (Start)

A063946(a(n)) = a(A063946(n)) = A117120(n), n > 0.

A065190(a(n)) = a(A065190(n)) = A092569(n), n > 0.

A258746(a(n)) = a(A258746(n)) = A165199(n), n > 0.

A258996(a(n)) = a(A258996(n)), n > 0.

A117120(a(n)) = a(A117120(n)), n > 0.

A092569(a(n)) = a(A092569(n)), n > 0. (End)

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x+1 and 1/x are in S. Then S is the set of positive rational numbers, which arise in generations as follows: g(1) = (1/1), g(2) = (1+1) = (2), g(3) = (2+1, 1/2) = (3/1, 1/2), g(4) = (4/1, 1/3, 3/2), ... . Once g(n-1) = (g(1), ..., g(z)) is defined, g(n) is formed from the vector (g(1) + 1, 1/g(1), g(2) + 1, 1/g(2), ..., g(z) + 1, 1/g(z)) by deleting all elements that are in a previous generation. A226080 is the sequence of denominators formed by concatenating the generations g(1), g(2), g(3), ... . It is easy to prove the following:

(1) Every positive rational is in S.

(2) The number of terms in g(n) is the n-th Fibonacci number, F(n) = A000045(n).

(3) For n > 2, g(n) consists of F(n-2) numbers < 1 and F(n-1) numbers > 1, hence the name "rabbit ordering" since the n-th generation has F(n-2) reproducing pairs and F(n-1) non-reproducing pairs, as in the classical rabbit-reproduction introduction to Fibonacci numbers.

(4) The positions of integers in S are the Fibonacci numbers.

(5) The positions of 1/2, 3/2, 5/2, ..., are Lucas numbers (A000032).

(6) Continuing from (4) and (5), suppose that n > 0 and 0 < r < n, where gcd(n,r) = 1. The positions in A226080 of the numbers congruent to r mod n comprise a row of the Wythoff array, W = A035513. The correspondence is sampled here:

row 1 of W: positions of n+1 for n>=0,

row 2 of W: positions of n+1/2,

row 3 of W: positions of n+1/3,

row 4 of W: positions of n+1/4,

row 5 of W: positions of n+2/3,

row 6 of W: positions of n+1/5,

row 7 of W: positions of n+3/4.

(7) If the numbers <=1 in S are replaced by 1 and those >1 by 0, the resulting sequence is the infinite Fibonacci word A003849 (except for the 0-offset first term).

(8) The numbers <=1 in S occupy positions -1 + A001950, where A001950 is the upper Wythoff sequence; those > 1 occupy positions given by -1 + A000201, where A000201 is the lower Wythoff sequence.

(9) The rules (1 is in S, and if x is in S, then 1/x and 1/(x+1) are in S) also generate all the positive rationals.

A variant which extends this idea to an ordering of all rationals is described in A226130. - M. F. Hasler, Jun 03 2013

The updown and downup zigzag limits are (-1 + sqrt(5))/2 and (1 + sqrt(5))/2; see A020651. - Clark Kimberling, Nov 10 2013

From Clark Kimberling, Jun 19 2014: (Start)

Following is a guide to related trees and sequences; for example, the tree A226080 is represented by (1, x+1, 1/x), meaning that 1 is in S, and if x is in S, then x+1 and 1/x are in S (except for x = 0).

All the positive integers:

A243571, A243572, A232559 (1, x+1, 2x)

A232561, A242365, A243572 (1, x+1, 3x)

A243573 (1, x+1, 4x)

All the integers:

A243610 (1, 2x, 1-x)

A232723, A242364

All the positive rationals:

A226080, A226081, A242359, A242360 (1, x+1, 1/x)

A243848, A243849, A243850 (1, x+1, 2/x)

A243851, A243852, A243853 (1, x+1, 3/x)

A243854, A243855, A243856 (1, x+1, 4/x)

A243574, A242308 (1, 1/x, 1/(x+1))

A241837, A243575 ({1,2,3}, x+4, 12/x)

A242361, A242363 (1, 1 + 1/x, 1/x)

A243613, A243614 (0, x+1, x/(x+1))

All the rationals:

A243611, A243612 (0, x+1, -1/(x+1))

A226130, A226131 (1, x+1, -1/x)

A243712, A243713 ({1,2,3}, x+1, 1/(x+1))

A243730, A243731 ({1,2,3,4}, x+1, 1/(x+1))

A243732, A243733 ({1,2,3,4,5}, x+1, 1/(x+1))

A243714, A243715

A243925, A243926, A243927 (1, x+1, -2/x)

A243928, A243929, A243930 (1, x+1, -3/x)

All the Gaussian integers:

A243924 (1, x+1, i*x)

All the Gaussian rational numbers:

A233694, A233695, A233696 (1, x+1, i*x, 1/x).

(End)

Also, a lexicographically minimal sequence of distinct positive integers such that a(n) is coprime to n. - Ivan Neretin, Apr 18 2015

The larger term of the pair (a(n), a(n+1)) is always odd. Had we started the sequence with a(1) = 0, it would be the lexicographically first sequence with this property if always extented with the smallest integer not yet present. - Eric Angelini, Feb 17 2017

From Yosu Yurramendi, Mar 21 2017: (Start)

This sequence is self-inverse. Except for the fixed point 1, it consists completely of 2-cycles: (2n, 2n+1), n > 0.

A020651(a(n)) = A020650(n), A020650(a(n)) = A020651(n), n > 0.

A245327(a(n)) = A245328(n), A245328(a(n)) = A245327(n), n > 0.

A063946(a(n)) = a(A063946(n)), n > 0.

A054429(a(n)) = a(A054429(n)) = A092569(n), n > 0.

A258996(a(n)) = a(A258996(n)), n > 0.

A258746(a(n)) = a(A258746(n)), n > 0. (End)

From Enrique Navarrete, Nov 13 2017: (Start)

With a(0)=0, and the rest of the sequence appended, a(n) is the smallest positive number not yet in the sequence such that the arithmetic mean of the first n+1 terms a(0), a(1), ..., a(n) is not an integer; i.e., the sequence is 0, 1, 3, 2, 5, 4, 7, 6, 9, 8, ...

Example: for n=5, (0 + 1 + 3 + 2 + 5)/5 is not an integer.

Fixed points are odd numbers >= 3 and also a(n) = n-2 for even n >= 4. (End)

The fractions are given in their reduced form, thus gcd(a(n), A020651(n)) = 1 for all n. - Antti Karttunen, May 26 2004

From Yosu Yurramendi, Jul 13 2014 : (Start)

If the terms (n>0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...

1,

2,1,

3,1,3,2,

4,1,4,3,5,2,5,3,

5,1,5,4,7,3,7,4, 7,2, 7,5, 8,3, 8,5,

6,1,6,5,9,4,9,5,10,3,10,7,11,4,11,7,9,2,9,7,12,5,12,7,11,3,11,8,13,5,13,8,

then the sum of the m-th row is 3^m (m = 0,1,2,), and each column is an arithmetic sequence.

If the rows are written in a right-aligned fashion:

1,

2,1,

3,1, 3,2,

4,1, 4,3, 5,2, 5,3,

5,1,5,4, 7,3, 7,4, 7,2, 7,5, 8,3, 8,5,

6,1,6,5,9,4,9,5,10,3,10,7,11,4,11,7,9,2,9,7,12,5,12,7,11,3,11,8,13,5,13,8,

each column k is a Fibonacci sequence. (End)

a(n2^m+1) = a(2n+1), n > 0, m > 0. - Yosu Yurramendi, Jun 04 2016

Thus a(n)/a(m) = d_1 + 1/(d_2 + 1/(d_3 + ... + 1/d_k)) where m = n - 2^floor(log_2(n)) + 1 and where d_j = b_j - b_(j+1) are the differences of the binary exponents b_j > b_(j+1) defined by: 4*n = 2^b_1 + 2^b_2 + 2^b_3 + ... 2^b_k.

All the rationals are uniquely represented by this sequence - compare Stern's diatomic sequence A002487.

This sequence lists the rationals >= 1 in order by the sum of the terms of their continued fraction expansions. For example, the numerators generated from partitions of 5 that do not end with 1 are listed together as 5, 7, 7, 8, 5, 7, 7, 8, since: 5/1 = [5]; 7/2 = [3;2]; 7/3 = [2;3]; 8/3 = [2;1,2]; 5/4 = [1;4]; 7/5 = [1;2,2]; 7/4 = [1;1,3]; 8/5 = [1;1,1,2].

From Yosu Yurramendi, Jun 23 2014: (Start)

If the terms (n>0) are written as an array:

1,

2,

3, 3,

4, 5, 4, 5,

5, 7, 7, 8, 5, 7, 7, 8,

6, 9,10,11, 9,12,11,13, 6, 9,10,11, 9,12,11,13,

7,11,13,14,13,17,15,18,11,16,17,19,14,19,18,21,7,11,13,14,13,17,15,18,11, ...

then the sum of the k-th row is 2*3^(k-2) for k>1, each column is an arithmetic progression. The differences of the arithmetic sequences give the sequence A071585 itself: a(2^(p+1)+k) - a(2^p+k) = a(k). A002487 and A007306 also have these properties. The first terms of columns, excluding a(0), give A086593.

If the rows (n>0) are written on right:

1;

2;

3, 3;

4, 5, 4, 5;

5, 7, 7, 8, 5, 7, 7, 8;

6, 9, 10, 11, 9, 12, 11, 13, 6, 9, 10, 11, 9, 12, 11, 13;

then each column is a Fibonacci sequence: a(2^(p+2)+k) = a(2^(p+1)+k) + a(2^p+k). The first terms of columns, excluding a(0), give A086593. (End)

n>1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (Franklin T. Adams-Watters's comment), that is the sequence obtained by adding numerator and denominator in the Calkin-Wilf enumeration system of positive rationals. A229742(n)/A071766(n) is also an enumeration system of all positive rationals (HCS system), and in each level m >= 0 (ranks between 2^m and 2^(m+1)-1) rationals are the same in both systems. Thus a(n) (A229742(n)+A071766(n)) has the same terms in each level as A007306. The same property occurs in all numerator+denominator sequences of enumeration systems of positive rationals, as, for example, A007306 (A007305+A047679), A086592 (A020650+A020651), and A268087 (A162909+A162910). - Yosu Yurramendi, Apr 06 2016

a(n) = A086592(A059893(n)), a(A059893(n)) = A086592(n), n > 0. - Yosu Yurramendi, May 30 2017

Form a tree of fractions by beginning with 1/1 and then giving every node i/j two descendants labeled i/(i+j) and j/(i+j).

The first row contains the single fraction 1/1,

the second row contains the single fraction 1/2,

and thereafter below each fraction i/j we write two fractions i/(i+j), j/(i+j).

If we just look at the numerators we recover the same sequence, and if we just look at the denominators we get A086593 with the terms (after the first) repeated.

Sequence A020651 is almost the same as this, except that it lacks one of the initial 1's, and the definition focuses on single numbers rather than pairs of numbers or fractions. For that reason it seems to be best to have a separate entry (this sequence) for the actual tree.