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%I #37 Dec 18 2021 22:20:57
%S 2,2,6,2,6,8,6,2,12,10,6,8,6,10,22,2,6,16,6,12,24,10,6,8,16,10,24,16,
%T 6,26,6,2,24,10,30,24,6,10,24,12,6,30,6,18,52,10,6,8,20,20,24,18,6,34,
%U 36,16,24,10,6,34,6,10,56,2,36,38,6,18,24,34,6,26,6,10,54,18,42,40,6,12
%N a(n) is twice the number of parts in all partitions of n into consecutive parts.
%C a(n) = 6 if and only if n is an odd prime.
%C a(n) = 2 if and only if n is a power of 2. - _Omar E. Pol_, Dec 13 2021
%H Antti Karttunen, <a href="/A338616/b338616.txt">Table of n, a(n) for n = 1..20000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = 2*A204217(n).
%e Illustration of initial terms:
%e Diagram
%e n a(n) _ _
%e 1 2 _|1 1|_
%e 2 2 _|1 _ _ 1|_
%e 3 6 _|1 |2 2| 1|_
%e 4 2 _|1 _| |_ 1|_
%e 5 6 _|1 |2 _ _ 2| 1|_
%e 6 8 _|1 _| |3 3| |_ 1|_
%e 7 6 _|1 |2 | | 2| 1|_
%e 8 2 _|1 _| _| |_ |_ 1|_
%e 9 12 _|1 |2 |3 _ _ 3| 2| 1|_
%e 10 10 _|1 _| | |4 4| | |_ 1|_
%e 11 6 _|1 |2 _| | | |_ 2| 1|_
%e 12 8 _|1 _| |3 | | 3| |_ 1|_
%e 13 6 _|1 |2 | _| |_ | 2| 1|_
%e 14 10 _|1 _| _| |4 _ _ 4| |_ |_ 1|_
%e 15 22 _|1 |2 |3 | |5 5| | 3| 2| 1|_
%e 16 2 |1 | | | | | | | | 1|
%e ...
%e a(n) is the total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the n-th level of the diagram.
%Y Cf. A054844 (twice the number of partitions of n into consecutive parts), A204217.
%Y Cf. A000079, A001227, A065091, A237591, A237593, A286001, A335616.
%K nonn
%O 1,1
%A _Omar E. Pol_, Nov 28 2020