A229940 Triangle T(n,k) read by rows: column k lists a periodic sequence with period 2*k, repeated: S, where S is a sequence formed by 2*k-1 1's together with a zero, starting in row A056220(k), n >=1, k >= 1.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1

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

Table of n, a(n) for n=1..96.

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

Index entries for sequences related to toothpick sequences

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

Columns 1-3: A059841, A166486, n >= 1, A097325, n >= 1.

Cf. A000005, A139250, A229942, A229950, A229951.

Sequence in context: A115788 A195062 A351077 * A179761 A317906 A102863

Adjacent sequences: A229937 A229938 A229939 * A229941 A229942 A229943

Keyword

nonn,tabf

Author

Omar E. Pol, Oct 04 2013

Status

approved