A267700 "Tree" sequence in a 90-degree sector of the cellular automaton of A160720.

0, 1, 2, 5, 6, 9, 12, 19, 20, 23, 26, 33, 36, 43, 50, 65, 66, 69, 72, 79, 82, 89, 96, 111, 114, 121, 128, 143, 150, 165, 180, 211, 212, 215, 218, 225, 228, 235, 242, 257, 260, 267, 274, 289, 296, 311, 326, 357, 360, 367, 374, 389, 396, 411, 426, 457, 464, 479, 494, 525, 540, 571, 602, 665, 666, 669, 672, 679, 682, 689

Offset

0,3

Comments

Conjecture: this is also the "tree" sequence in a 120-degree sector of the cellular automaton of A266532.

It appears that this is also the partial sums of A038573.

a(n) is also the total number of ON cells after n-th stage in the tree that arises from one of the four spokes in a 90-degree sector of the cellular automaton A160720 on the square grid.

Note that the structure of A160720 is also the "outward" version of the Ulam-Warburton cellular automaton of A147562.

It appears that A038573 gives the number of cells turned ON at n-th stage.

Conjecture: a(n) is also the total number of Y-toothpicks after n-th stage in the tree that arises from one of the three spokes in a 120-degree sector of the cellular automaton of A266532 on the triangular grid.

Note that the structure of A266532 is also the "outward" version of the Y-toothpick cellular automaton of A160120.

It appears that A038573 also gives the number of Y-toothpicks added at n-th stage.

Comment from N. J. A. Sloane, Jan 23 2016: All the above conjectures are true!

From Gus Wiseman, Mar 31 2019: (Start)

a(n) is also the number of nondecreasing binary-containment pairs of positive integers up to n. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second. For example, the a(1) = 1 through a(6) = 12 pairs are:

(1,1) (1,1) (1,1) (1,1) (1,1) (1,1)

(2,2) (1,3) (1,3) (1,3) (1,3)

(2,2) (2,2) (1,5) (1,5)

(2,3) (2,3) (2,2) (2,2)

(3,3) (3,3) (2,3) (2,3)

(4,4) (3,3) (2,6)

(4,4) (3,3)

(4,5) (4,4)

(5,5) (4,5)

(4,6)

(5,5)

(6,6)

(End)

Links

Table of n, a(n) for n=0..69.

David Applegate, The movie version

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to cellular automata

Index entries for sequences related to toothpick sequences

Formula

a(n) = (A160720(n+1) - 1)/4.

Conjecture 1: a(n) = (A266532(n+1) - 1)/3.

Conjecture 2: a(n) = A160720(n+1) - A266532(n+1).

All of the above conjectures are true. - N. J. A. Sloane, Jan 23 2016

(Conjecture) a(n) = A267610(n) + n. - Gus Wiseman, Mar 31 2019

Mathematica

Accumulate[Table[2^DigitCount[n, 2, 1]-1, {n, 0, 30}]] (* based on conjecture confirmed by Sloane, Gus Wiseman, Mar 31 2019 *)

Crossrefs

Cf. A000120, A038573, A147562, A160120, A160720, A161336, A169779, A266532, A266534, A266536.

Cf. A006218, A019565, A070939, A080572, A267610, A267700.

Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325124.

Sequence in context: A046962 A022486 A349895 * A161910 A151920 A160714

Adjacent sequences: A267697 A267698 A267699 * A267701 A267702 A267703

Keyword

nonn

Author

Omar E. Pol, Jan 19 2016

Status

approved