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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.
7
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.
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Illustration of a graph in which every set formed by j toothpicks connected by their endpoints has been replaced by an edge of length j: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.
Sequence in context: A115788 A195062 A351077 * A373157 A179761 A317906
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Oct 04 2013
STATUS
approved