cp's OEIS Frontend

Essentially the same sequence as A001316, which has much more information, and also A159913. - N. J. A. Sloane, Jun 05 2009

Smallest number with same number of 1's in its binary expansion as n.

Fixed point of the morphism 0 -> 01, 1 -> 13, 3 -> 37, ... = k -> k, 2k+1, ... starting from a(0) = 0; 1 -> 01 -> 0113 -> 01131337 -> 011313371337377(15) -> ..., . - Robert G. Wilson v, Jan 24 2006

From Gary W. Adamson, Jun 04 2009: (Start)

As an infinite string, 2^n terms per row starting with "1": (1; 1,3; 1,3,3,7; 1,3,3,7,3,7,7,15; 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31;...)

Row sums of that triangle = A027649: (1, 4, 14, 46, 454, ...); where the next row sum = current term of A027649 + next term in finite difference row of A027649, i.e., (1, 3, 10, 32, 100, 308, ...) = A053581. (End)

From Omar E. Pol, Jan 24 2016: (Start)

Partial sums give A267700.

a(n) is also the number of cells turned ON at n-th generation of the cellular automaton of A267700 in a 90-degree sector on the square grid.

a(n) is also the number of Y-toothpicks added at n-th generation of the structure of A267700 in a 120-degree sector on the triangular grid. (End)

Row sums of A090971. - Nikolaos Pantelidis, Nov 23 2022

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)

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

A pair of positive integers is divisible if the first divides the second, and 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.

We work on the vertices of the square grid Z^2, and define the neighbors of a cell to be the four closest cells along the diagonals.

We start at stage 0 with all cells in OFF state.

At stage 1, we turn ON a single cell at the origin.

Once a cell is ON it stays ON.

At each subsequent stage, a cell in turned ON if exactly one of its neighboring cells that are no further from the origin is ON.

The "no further from the origin" condition matters for the first time at stage 8, when only A160721(8) = 28 cells are turned ON, and a(8) = 77. In contrast, A147562(8) = 85, A147582(8) = 36.

This CA also arises as the cross-section in the (X,Y)-plane of the CA in A151776.

In other words, a cell is turned ON if exactly one of its vertices touches an exposed vertex of a ON cell of the previous generation. A special rule for this sequence is that every ON cell has only one vertex that should be considered not exposed: its nearest vertex to the center of the structure.

Analog to the "outward" version (A266532) of the Y-toothpick cellular automaton of A160120 on the triangular grid, but here we have ON cells on the square grid. See also the formula section. - Omar E. Pol, Jan 19 2016

This cellular automaton can be interpreted as the outward version of the Ulam-Warburton two-dimensional cellular automaton (see A147562). - Omar E. Pol, Jun 22 2017

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

The version for ordered pairs is A080572.

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

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.