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

Author

Omar E. Pol, Jan 19 2016

Keywords

• nonn

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

• 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

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.

Programs

Mathematica

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

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