cp's OEIS Frontend

The original definition was: "A threequence, a 3-way partitioning of the integers: define a,b,c,A,B,C by a(0)=true, b(0)=c(0)=A(0)=B(0)=C(0)=false, a(n)=a(m) OR C(m) OR B(m), b(n)= b(m) OR A(m) OR C(m), c(n)= c(m) OR B(m) OR A(m), A(n)= A(m) OR b(m) OR c(m), B(n)= B(m) OR c(m) OR a(m), C(n)= C(m) OR a(m) OR b(m), where m = [ (n+1)/3 ]; sequence gives n such that a(2n) is true." - Bradley Klee, Apr 16 2019

The original definition was: "A threequence, a 3-way partitioning of the integers: define a,b,c,A,B,C by a(0)=true, b(0)=c(0)=A(0)=B(0)=C(0)=false, a(n)=a(m) OR C(m) OR B(m), b(n)= b(m) OR A(m) OR C(m), c(n)= c(m) OR B(m) OR A(m), A(n)= A(m) OR b(m) OR c(m), B(n)= B(m) OR c(m) OR a(m), C(n)= C(m) OR a(m) OR b(m), where m = [ (n+1)/3 ]; sequence gives n such that b(2n) is true." - Sean A. Irvine, Apr 30 2019

The original definition was: "A threequence, a 3-way partitioning of the integers: define a,b,c,A,B,C by a(0)=true, b(0)=c(0)=A(0)=B(0)=C(0)=false, a(n)=a(m) OR C(m) OR B(m), b(n)= b(m) OR A(m) OR C(m), c(n)= c(m) OR B(m) OR A(m), A(n)= A(m) OR b(m) OR c(m), B(n)= B(m) OR c(m) OR a(m), C(n)= C(m) OR a(m) OR b(m), where m = [ (n+1)/3 ]; sequence gives n such that c(2n) is true." - Sean A. Irvine, Apr 30 2019

Start with 0 and apply the morphism 0->011 and 1->010 repeatedly.

This sequence draws the Sierpinski gasket, when iterating the following odd-even drawing rule: If "1" then draw a segment forward, if "0" then draw a segment forward and turn 120 degrees right if in odd position or left if in even position.

From Dimitri Hendriks, Jun 29 2010: (Start)

This sequence is the first difference of the Mephisto Waltz A064990, i.e., a(n) = A064990(n) + A064990(n+1), where '+' is addition modulo 2.

This sequence can also be generated as a Toeplitz word: First consider the periodic word 0,1,$,0,1,$,0,1,$,... and then fill the gaps $ by the bitwise negation of the sequence itself: 0,1,1,0,1,0,0,1,0,.... See the Allouche/Bacher reference for a precise definition of Toeplitz sequences. (End)

From Joerg Arndt, Jan 21 2013: (Start)

Identical to the morphism 0-> 011010010, 1->011010011 given on p. 100 of the Fxtbook (see link), because 0 -> 011 -> 011010010 and 1 -> 010 -> 011010011.

This sequence gives the turns (by 120 degrees) of the R9-dragon curve (displayed on p. 101) which can be rendered as follows:

[Init] Set n=0 and direction=0.

[Draw] Draw a unit line (in the current direction). Turn left/right if a(n) is zero/nonzero respectively.

[Next] Set n=n+1 and goto (draw).

(End)

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.

Defines a function on all the integers, but only nonnegative terms are in the data. A147991 gives the nonnegative fixed points of the function and the nonnegative part of its image.

Consider a Sierpinski arrowhead curve to be formed of vectors placed head to tail and numbered consecutively from 0 at its axis of symmetry. Vector a(n) equals vector n.

Removing all 0's from the balanced ternary expansion of n yields a(n). - Charlie Neder, Jun 03 2019

Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:

Y

/

/

0 ---- X

The Sierpinski arrowhead curve can be represented using an L-system.

This sequence and A036581 are squarefree (they do not contain any substring XX).

Appears to be A307672(n) mod 3. - Peter Munn, Aug 22 2021