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A340848 a(n) is the number of edges in the diagram of the symmetric representation of sigma(n) with subparts. 3
4, 6, 8, 10, 10, 14, 12, 14, 16, 16, 14, 24, 14, 18, 24, 22, 16, 28, 16, 26, 26, 22, 18, 36, 24, 22, 28, 30, 20, 44, 20, 30 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Since the diagram is symmetric so all terms are even numbers.
For another version see A340846 from which first differs at a(6).
For the definition of subparts see A279387. For more information about the subparts see also A237271, A280850, A280851, A296508, A335616.
Note that in this version of the diagram of the symmetric representation of sigma(n) all regions are called "subparts". The number of subparts equals A001227(n).
LINKS
FORMULA
a(n) = A340847(n) + A001227(n) - 1 (Euler's formula).
EXAMPLE
Illustration of initial terms:
. _ _ _ _
. _ _ _ |_ _ _ |_
. _ _ _ |_ _ _| | |_|_
. _ _ |_ _ |_ |_ _ |_ _ |
. _ _ |_ _|_ |_ | | | | |
. _ |_ | | | | | | | | |
. |_| |_| |_| |_| |_| |_|
.
n: 1 2 3 4 5 6
a(n): 4 6 8 10 10 14
.
For n = 6 the diagram has 14 edges so a(6) = 14.
On the other hand the diagram has 13 vertices and two subparts or regions, so applying Euler's formula we have that a(6) = 13 + 2 - 1 = 14.
. _ _ _ _ _
. _ _ _ _ _ |_ _ _ _ _|
. _ _ _ _ |_ _ _ _ | |_ _
. |_ _ _ _| | |_ |_ |
. |_ |_ |_ _ |_|_ _
. |_ _ |_ _ | | |
. | | | | | |
. | | | | | |
. | | | | | |
. |_| |_| |_|
.
n: 7 8 9
a(n): 12 14 16
.
For n = 9 the diagram has 16 edges so a(9) = 16.
On the other hand the diagram has 14 vertices and three subparts or regions, so applying Euler's formula we have that a(9) = 14 + 3 - 1 = 16.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
. n a(n) Diagram
--------------------------------------------------------------------------
_
1 4 |_| _
_| | _
2 6 |_ _| | | _
_ _|_| | | _
3 8 |_ _| _| | | | _
_ _| _| | | | | _
4 10 |_ _ _| _|_| | | | | _
_ _ _| _ _| | | | | | _
5 10 |_ _ _| | _ _| | | | | | | _
_ _ _| |_| _|_| | | | | | | _
6 14 |_ _ _ _| _| _ _| | | | | | | | _
_ _ _ _| _| _ _| | | | | | | | | _
7 12 |_ _ _ _| | _| _ _|_| | | | | | | | | _
_ _ _ _| | _| | _ _| | | | | | | | | | _
8 14 |_ _ _ _ _| |_ _| | _ _| | | | | | | | | | | _
_ _ _ _ _| _ _|_| _ _|_| | | | | | | | | | |
9 16 |_ _ _ _ _| | _| _| _ _ _| | | | | | | | | |
_ _ _ _ _| | _| _| _ _ _| | | | | | | | |
10 16 |_ _ _ _ _ _| | _| _| | _ _|_| | | | | | |
_ _ _ _ _ _| | _| _| | _ _ _| | | | | |
11 14 |_ _ _ _ _ _| | |_ _| _| | _ _ _| | | | |
_ _ _ _ _ _| | _ _| _|_| _ _ _|_| | |
12 24 |_ _ _ _ _ _ _| | _ _| _ _| | _ _ _| |
_ _ _ _ _ _ _| | _| | _ _| | _ _ _|
13 14 |_ _ _ _ _ _ _| | | _| |_| _| |
_ _ _ _ _ _ _| | |_ _| _| _|
14 18 |_ _ _ _ _ _ _ _| | _ _| _|
_ _ _ _ _ _ _ _| | _ _|
15 24 |_ _ _ _ _ _ _ _| | |
_ _ _ _ _ _ _ _| |
16 22 |_ _ _ _ _ _ _ _ _|
...
CROSSREFS
Cf. A001227 (number of subparts or regions).
Cf. A340847 (number of vertices).
Cf. A340846 (number of edges in the diagram only with parts).
Cf. A317292 (total number of edges in the unified diagram).
Sequence in context: A340846 A167146 A020891 * A090967 A349707 A272475
KEYWORD
nonn,more
AUTHOR
Omar E. Pol, Jan 24 2021
STATUS
approved

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Last modified May 26 08:22 EDT 2024. Contains 372813 sequences. (Running on oeis4.)