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%I #25 Nov 16 2013 13:37:53
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv13.jpg">Illustration of initial terms of the divisor function (A000005)</a>, see the third picture.
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e Illustration of initial terms:
%e -----------------------------------------
%e Triangle Diagram A000005
%e -----------------------------------------
%e . 1
%e 1; |
%e 0; 2
%e 1; |
%e 0; 2
%e 1; |
%e 0; 3
%e 1, 1; | |
%e 0, 1; | 2
%e 1, 1; | |
%e 0, 0; 4
%e 1, 1; | |
%e 0, 1; | 2
%e 1, 1; | |
%e 0, 0; 4
%e 1, 1; | |
%e 0, 1; | 3
%e 1, 1, 1; | | |
%e 0, 0, 1; | 4
%e 1, 1, 1; | | |
%e 0, 1, 1; | | 2
%e 1, 1, 1; | | |
%e 0, 0, 0; 6
%e 1, 1, 1; | | |
%e 0, 1, 1; | | 2
%e 1, 1, 1; | | |
%e 0, 0, 1; | 4
%e 1, 1, 1; | | |
%e 0, 1, 0; | 4
%e 1, 1, 1; | | |
%e 0, 0, 1; | 5
%e 1, 1, 1, 1; | | | |
%e ...
%e Illustration of the structure with the toothpicks in horizontal direction. The number of exposed endpoints in the even-indexed column gives A000005:
%e . -|
%e . ------- ------- ---|
%e . ----- ----- ----- ----- ----- ---|
%e . --- --- --- --- --- --- --- --- --- --- ---|
%e . - - - - - - - - - - - - - - - - - - - - - - - - -|
%e . 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 |
%e .
%e Illustration of a graph in which every set formed by j toothpicks connected by their endpoints has been replaced by an edge of length j:
%e . _|
%e . _______ _______ ___|
%e . _____ _____ _____ _____ _____ ___|
%e . ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___|
%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e . |
%e . 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 |
%e .
%e See also Links section.
%Y Columns 1-3: A059841, A166486, n >= 1, A097325, n >= 1.
%Y Cf. A000005, A139250, A229942, A229950, A229951.
%K nonn,tabf
%O 1
%A _Omar E. Pol_, Oct 04 2013
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