cp's OEIS Frontend

Numbers whose set of base 4 digits is {0,3}. - Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 4 for every i. - Ray Chandler, Aug 03 2004

The first 2^n terms of the sequence could be obtained using the Cantor-like process for the segment [0, 4^n-1]. E.g., for n=1 we have [0, {1, 2}, 3] such that numbers outside of braces are the first 2 terms of the sequence; for n=2 we have [0, {1, 2}, 3, {4, 5, 6, 7, 8, 9, 10, 11}, 12, {13, 14}, 15] such that the numbers outside of braces are the first 4 terms of the sequence, etc. - Vladimir Shevelev, Dec 17 2012

From Emeric Deutsch, Jan 26 2018: (Start)

Also, the indices of the compositions having only even parts. For the definition of the index of a composition, see A298644. For example, 195 is in the sequence since its binary form is 11000011 and the composition [2,4,2] has only even parts. 132 is not in the sequence since its binary form is 10000100 and the composition [1,4,1,2] also has odd parts.

The command c(n) from the Maple program yields the composition having index n. (End)

After the k-th step of generating the Koch snowflake curve, label the edges of the curve consecutively 0..3*4^k-1 starting from a vertex of the originating triangle. a(0), a(1), ... a(2^k-1) are the labels of the edges contained in one edge of the originating triangle. Add 4^k to each label to get the labels for the next edge of the triangle. Compare with A191108 in respect of the Sierpinski arrowhead curve. - Peter Munn, Aug 18 2019

Positive numbers that can be written in balanced ternary without a 0 trit. - J. Hufford, Jun 30 2015

Let S be the set of terms. Define c: Z -> P(R) so that c(m) is the translated Cantor ternary set spanning [m-0.5, m+0.5], and let C be the union of c(m) for all m in S U {0} U -S. C is the closure of the translated Cantor ternary set spanning [-0.5, 0.5] under multiplication by 3. - Peter Munn, Jan 31 2022

Related sequences for various choices of i and k as defined in A190803:

A003278: (i,k) = (-2,-1)

A191106: (i,k) = (-2, 0)

A191107: (i,k) = (-2, 1)

A191108: (i,k) = (-2, 2)

A153775: (i,k) = (-1, 0)

A147991: (i,k) = (-1, 1)

A191109: (i,k) = (-1, 2)

A005836: (i,k) = ( 0, 1)

A191110: (i,k) = ( 0, 2)

A132140: (i,k) = ( 1, 2)

For a=A191106, we have closure properties: the integers in (2+a)/3 comprise a; the integers in a/3 comprise a.

For k >= 1, m = a(i), 1 <= i <= 2^k seems to be m such that m/(3^k+1) is in the Cantor set (except that m = 0 and m = 3^k+1 do not appear). For k >= 2, m = (a(i)-1)/2, 1 <= i <= 2^k seems to be m such that m/((3^k-1)/2) is in the Cantor set. - Peter Munn, Jul 06 2019

Every even number is the sum of two (possibly equal) terms. More specifically: terms a(1) through a(2^n) = 3^n sum to even numbers 2 times 1 through 3^n. Every even number is infinitely often the difference of two terms. Since the sequence is equal to 2*A005836(n) + 1, these properties follow immediately from similar properties of A005836 for every number. - Aad Thoen, Feb 17 2022

if A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+2*3^n), similar to A003278. - Arie Bos, Jul 26 2022

The adjective "balanced" indicates that the fixed point a(0)=0 descends through iteration along a central dividing line, which bisects the ternary family tree into left and right halves, equal by node cardinality (see examples). From the original Gosper and Ziegler-Hunts reference (see links), a(k) = d(k) mod 6. The function d(k) draws left and right halves of the Sierpiński Arrowhead curve (see links). Alphanumeric transformation {0->a, 2->b, 4->c, 3->A, 5->B, 1->C} obtains d(k) in the form of lettered sets. By design, letters {a,b,c} occur only on even indices, while letters {A,B,C} occur only on odd indices. According to the principal eigenvector of the substitution system, occurrence tallies should asymptotically approach a uniform distribution over the six numbers or letters.

From Peter Munn, May 28 2019: (Start)

The sequence maps to half of an infinite Sierpinski arrowhead curve by mapping the values 0..5 to six unit vectors spaced at equal angles (Pi/3) in counterclockwise (or clockwise) order, then placing the vector image of each sequence term head to tail. Curve edges indexed 0..121 form the upper half of the curve in Figure 5 of the Gosper & Ziegler-Hunts reference (see links). The figure has the vector image of 0 pointing upwards, the red-colored segment runs from index -40 to +40 and the blue-colored segment from 41 to 121.

The arrowhead curve (both halves and continued to infinity) will align with an infinite Sierpinski gasket so that each of its edges is contained in the boundary of either the plane sector occupied by the gasket or a triangular region of the gasket's complement. Every length 3 segment of these boundaries contains exactly one edge of the arrowhead curve. See the link for the aligned curves.

For a given triangular boundary (or a given edge of the gasket sector boundary) the indices of the arrowhead edges it contains differ by multiples of 4. The edges in the gasket sector boundary are listed (absolute value of index) in A191108. Otherwise, edge n seems to be contained in a triangular boundary of side 2^(A307744(n)-1).

The arrowhead curve divides the plane into two regions. Denote the region that is wholly within the sector occupied by the gasket as the inside of the arrowhead. The curve's even-indexed edges are in triangular boundaries that lie inside the arrowhead, and the odd-indexed edges are not.

When the term-to-vector map described above is applied to the sequence bisections, we get related curves. The even-indexed curve reproduces the boundaries of all triangular regions of the gasket's complement, of unit side and greater, that lie inside the arrowhead; the odd-indexed curve reproduces the boundaries of the equivalent regions outside the arrowhead plus the gasket sector boundary. See the link for the aligned curves.

Recall that every length 3 boundary segment contains exactly one arrowhead edge. In the curve drawn by a(0), a(2), a(4), ... the image of a(6n) co-incides with the image of a(2n) in the arrowhead curve, and the images of a(6n-2), a(6n) and (6n+2) form a length 3 boundary segment. Similarly, in the curve drawn by a(1), a(3), a(5), ... the image of a(6n+3) co-incides with the image of a(2n+1) in the arrowhead curve, and the images of a(6n+1), a(6n+3) and a(6n+5) form a length 3 boundary segment.

One bisection produces vectors that draw triangular boundaries clockwise, the other counterclockwise. This must be so, because (1) the full sequence alternates odd and even, (2) opposite vectors are images of numbers with opposite parity, and (3) the gasket complement's triangular regions have the same orientation.

(End)

Nonnegative integers whose ternary representation contains only digits 0 and 2 except for at most a single digit 1 that is followed only by 0's.

Nonnegative integers that can be written in base 3 using only 0's and 2's, allowing the use of the "decimal" point (.) and replacing ....10..0(.) by ....02..2(.)2222...

Note that fractions are not reduced.

List of integers in the closure of the ternary Cantor set under multiplication by 3. The closure is the union of the translated ternary Cantor sets spanning [a(1), a(2)], [a(3), a(4)], [a(5), a(6)], ... . - Peter Munn, Jul 09 2019

The sequence extends to negative n by defining a(n) = a(-n).

For k >= 1 numbers 1..k occur with the same periodic and mirror symmetries as in ruler function A051064, in which k occurs 3 times more frequently than k+1. Here k occurs 3/2 times more frequently than k+1, precisely 2^(k-1) times in every 3^k terms. 0 has asymptotic density 0. Taking a trisection shows some scale symmetry, again comparable to ruler functions, as illustrated in the example section.

The links include a pin plot of a(0..162) aligned above an inverted plot of A051064 (the emphatic marking of 0's is significant). Between each n_k where A051064(n_k) = k >= 2 and the nearest n_k' where A051064(n_k') > k (or n_k' = 0 if nearer), there are 2^(k-2) indices where k occurs in this sequence, forming a 2^(k-2)-tuple. The 2^(k-2)-tuples have identical patterns and each has symmetry about an n_(k-1) where A051064(n_(k-1)) = k-1.

For a given k, the tuples described above are periodic with two per fundamental period, and the closest pairs of these tuples jointly form the pattern of one of the equivalent tuples for k+1. These patterns relate to the nonperiodic pattern for 0's and to the Cantor set as follows.

Let S_k be the sequence of positive indices at which k occurs, with 3^(k-2) subtracted when k >= 2. Given its ruler-type symmetries, S_k k >= 2 is determined by its first 2^(k-2) terms, which are the same as the first 2^(k-2) terms of S_i for i > k. The limiting sequence as k goes to infinity is S_0, which is A191108. {A191108(i)/(2*3^k) | 1 <= i <= 2^k} is the set of center points of the intervals deleted at step k+1 when generating the Cantor ternary set. This leads to the following scaling property.

Define c: Z -> P(R) so that c(n) is the scaled and translated Cantor ternary set spanning [n-1, n+1], and let C_k be the union of c(n) for all integer n with a(n) = k. Clearly C_1 consists of a scaled Cantor set repeated with period 3. (The set's two nonempty thirds occur at alternating intervals of 4/3 and 5/3.) For k >= 1, C_k is C_1 scaled by 3^(k-1), consisting therefore of a scaled Cantor set repeated with period 3^k. C_0 is special: C_0 = (C_0)*3 = (C_0)/3 = -C_0. Specifically, (C_0)/2 is the closure of the Cantor ternary set under multiplication by 3 and by -1.

Take a Sierpinski arrowhead curve formed of unit edges numbered consecutively from 0 at its axis of symmetry and aligned with an infinite Sierpinski gasket so that each edge is contained in the boundary of either the plane sector occupied by the gasket or a triangular region of the gasket's complement. If a(n) = 0, the n-th edge is contained in the sector boundary, otherwise the relevant triangular region seems to have side 2^(a(n)-1). See A307672 for a fuller description. The conjectured formulas below (that use A094373) derive from summing areas of regions within the gasket. - Corrected by Peter Munn, Aug 09 2019

From Charlie Neder, Jul 05 2019: (Start)

For each n, define the "2-balanced ternary expansion" E(n) as follows:

- E(n) begins with 0 or 1, according to the parity of n.

- The following digits are +, 0, or - as in standard balanced ternary, except + and - correspond to +2 and -2, respectively.

For example, we have E(4) = 0+-, E(7) = 10-, and E(13) = 1+-.

Then a(n) is the distance from the end of the rightmost 0, counting the last digit as 1, or 0 if 0 never appears. (End)

The sequence extends to negative n by defining a(n) = a(-n).

Consider a Sierpinski arrowhead curve formed of unit edges numbered consecutively from 0 at its axis of symmetry and aligned with an infinite Sierpinski gasket so that each edge is contained in the boundary of either the plane sector occupied by the gasket or a triangular region of the gasket's complement. If a(n) = 0, the n-th edge is contained in the sector boundary, otherwise the relevant triangular region has side a(n). Every length 3 segment of these boundaries contains exactly one edge of the arrowhead curve. A191108 lists positive n such that edge n is contained in the plane sector boundary. See A307672 for further properties.

See the graphic (in the links) of the arrowhead curve aligned with the gasket. Note the even-indexed edges (colored red) are the edges contained in a triangular region boundary on the left side of the vector. - Peter Munn, Jul 29 2019

a(n) = 0 if A307744(n) = 0, otherwise a(n) = 2^(A307744(n) - 1). Thus, this sequence has the symmetries of A307744, which are similar to ruler functions (especially A051064) and described further in A307744.

When listening to this, set pitch modulus to 35 or 36.

At the (k+1)-th step of generating the Cantor set, the offsets from 1/4 to the center points of the deleted middle thirds are {a(i)/(4*(-3)^k) : 1 <= i <= 2^k}. Clearly, these offsets are negated for use with respect to 3/4.

Note that each quarter point of the Cantor ternary set, C, is also a quarter point of an interval-constrained subset of C that is an image of C scaled by 3^(-k) for all k >= 1.

If we replace -3m-4 and -3m+4 in the definition with -3m-2 and -3m+2 we get the terms of A191108 and their negation.