OFFSET
1
COMMENTS
The sequence is connected with the divisor function A000005. Consider a structure in which the 1's of the triangle are replaced by toothpicks of length 1 as shown in the diagram in the Example section. Note that in every column the successive toothpicks are connected by their endpoints. The number of exposed endpoints that touch row 2*n - 2 of the structure is equal to A000005(n), the number of divisors of n, with n >= 1.
LINKS
Omar E. Pol, Illustration of initial terms of the divisor function (A000005), see the third picture.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
EXAMPLE
Illustration of initial terms:
-----------------------------------------
Triangle Diagram A000005
-----------------------------------------
. 1
1; |
0; 2
1; |
0; 2
1; |
0; 3
1, 1; | |
0, 1; | 2
1, 1; | |
0, 0; 4
1, 1; | |
0, 1; | 2
1, 1; | |
0, 0; 4
1, 1; | |
0, 1; | 3
1, 1, 1; | | |
0, 0, 1; | 4
1, 1, 1; | | |
0, 1, 1; | | 2
1, 1, 1; | | |
0, 0, 0; 6
1, 1, 1; | | |
0, 1, 1; | | 2
1, 1, 1; | | |
0, 0, 1; | 4
1, 1, 1; | | |
0, 1, 0; | 4
1, 1, 1; | | |
0, 0, 1; | 5
1, 1, 1, 1; | | | |
...
Illustration of the structure with the toothpicks in horizontal direction. The number of exposed endpoints in the even-indexed column gives A000005:
. -|
. ------- ------- ---|
. ----- ----- ----- ----- ----- ---|
. --- --- --- --- --- --- --- --- --- --- ---|
. - - - - - - - - - - - - - - - - - - - - - - - - -|
. 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 |
.
Illustration of a graph in which every set formed by j toothpicks connected by their endpoints has been replaced by an edge of length j:
. _|
. _______ _______ ___|
. _____ _____ _____ _____ _____ ___|
. ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___|
. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
. |
. 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 |
.
See also Links section.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Oct 04 2013
STATUS
approved