cp's OEIS Frontend

From Gus Wiseman, Mar 30 2019: (Start)

It appears that a(n) is also the number of increasing binary-containment pairs of distinct positive integers up to n + 1. 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(2) = 2 through a(8) = 14 pairs are:

{1,3} {1,3} {1,3} {1,3} {1,3} {1,3} {1,3}

{2,3} {2,3} {1,5} {1,5} {1,5} {1,5} {1,5}

{2,3} {2,3} {1,7} {1,7} {1,7}

{4,5} {2,6} {2,3} {2,3} {1,9}

{4,5} {2,6} {2,6} {2,3}

{4,6} {2,7} {2,7} {2,6}

{3,7} {3,7} {2,7}

{4,5} {4,5} {3,7}

{4,6} {4,6} {4,5}

{4,7} {4,7} {4,6}

{5,7} {5,7} {4,7}

{6,7} {6,7} {5,7}

{6,7}

{8,9}

(End)

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)

Conjectured to be less than or equal to lcs(n) (see sequence A063437). The value of a(2^n) is that given in Stinson and van Rees and the value of a(2^n-1) is that given in Fu, Fu and Liao. This function gives an easy way to generate these two constructions.

From Gus Wiseman, Mar 30 2019: (Start)

Also the number of ordered pairs of positive integers up to n with at least one binary carry. A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the a(2) = 1 through a(6) = 15 ordered pairs are:

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

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

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

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

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

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

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

(4,4) (3,3)

(3,5)

(4,4)

(4,5)

(5,1)

(5,3)

(5,4)

(5,5)

(End)

a(n) is also the number of even elements in the n X n symmetric Pascal matrix. - Stefano Spezia, Nov 14 2022

Two positive integers are divisible if the first divides the second, and they have a binary carry if the positions of 1's in their reversed binary expansion overlap.

a(2k+1) = a(2k), since an odd number and any divisor will overlap in the last digit. Additionally, a(2k+2) > a(2k+1) because the pair {1,2k+2} is always valid. Therefore, every term appears exactly twice. - Charlie Neder, Apr 02 2019

Two positive integers are divisible if the first divides the second, and have a binary carry if the positions of 1's in their reversed binary expansion overlap.