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A317292
a(n) is the total number of edges after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 0.
3
0, 4, 8, 14, 20, 26, 36, 42, 50, 60, 70, 76, 92, 98, 108, 124, 136, 142, 160, 166, 182, 198, 208, 214, 238, 250, 260, 276, 294, 300
OFFSET
0,2
COMMENTS
All terms are even numbers.
Note that in the diagram the number of regions or subparts equals A060831, the partial sums of A001227, n >= 1.
FORMULA
a(n) = A317293(n) + A060831(n) - 1 (Euler's formula).
EXAMPLE
Illustration of initial terms (n = 1..9):
. _ _ _ _
. _ _ _ |_ _ _ |_
. _ _ _ |_ _ _| |_ _ _| |_|_
. _ _ |_ _ |_ |_ _ |_ _ |_ _ |_ _ |
. _ _ |_ _|_ |_ _|_ | |_ _|_ | | |_ _|_ | | |
. _ |_ | |_ | | |_ | | | |_ | | | | |_ | | | | |
. |_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 4 8 14 20 26 36
.
. _ _ _ _ _
. _ _ _ _ _ |_ _ _ _ _|
. _ _ _ _ |_ _ _ _ | |_ _ _ _ |_ _
. |_ _ _ _| |_ _ _ _| |_ |_ _ _ _| |_ |
. |_ _ _ |_ |_ _ _ |_ |_ _ |_ _ _ |_ |_|_ _
. |_ _ _| |_|_ _ |_ _ _| |_|_ _ | |_ _ _| |_|_ _ | |
. |_ _ |_ _ | | |_ _ |_ _ | | | |_ _ |_ _ | | | |
. |_ _|_ | | | | |_ _|_ | | | | | |_ _|_ | | | | | |
. |_ | | | | | | |_ | | | | | | | |_ | | | | | | | |
. |_|_|_|_|_|_|_| |_|_|_|_|_|_|_|_| |_|_|_|_|_|_|_|_|_|
.
. 42 50 60
.
.
Illustration of the two-dimensional diagram after 29 stages (contains 300 edges, 239 vertices and 62 regions or subparts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _ | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _ | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _ |
|_ _ _ _ _ _ _ _ _ _ _ | | |_ _ | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_ |_
|_ _ _ _ _ _ _ _ _ _ | | |_ _ |_ _| |_|_
|_ _ _ _ _ _ _ _ _ _| | |_ _ | |_ |_ _ |_ _
|_ _ _ _ _ _ _ _ _ | |_ _ _| |_ |_ | |_ _ |
|_ _ _ _ _ _ _ _ _| | |_ _ |_ |_ |_|_ _ | |
|_ _ _ _ _ _ _ _ | |_ _ |_ _|_ |_ _ | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| | |_ _| |_ | |_ _ | | |_|_ _ _ _ _ | |
|_ _ _ _ _ _ _ | |_ _ |_ |_|_ | | |_|_ _ _ _ _ | | | |
|_ _ _ _ _ _ _| |_ _ |_ |_ _ | | |_ _ _ _ _ | | | | | |
|_ _ _ _ _ _ | |_ |_ |_ | | |_|_ _ _ _ | | | | | | | |
|_ _ _ _ _ _| |_ _| |_|_ | |_|_ _ _ _ | | | | | | | | | |
|_ _ _ _ _ | |_ |_ _ | |_ _ _ _ | | | | | | | | | | | |
|_ _ _ _ _| |_ |_ | |_|_ _ _ | | | | | | | | | | | | | |
|_ _ _ _ |_ _|_ |_|_ _ _ | | | | | | | | | | | | | | | |
|_ _ _ _| |_ | |_ _ _ | | | | | | | | | | | | | | | | | |
|_ _ _ |_ |_|_ _ | | | | | | | | | | | | | | | | | | | |
|_ _ _| |_|_ _ | | | | | | | | | | | | | | | | | | | | | |
|_ _ |_ _ | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_ | | | | | | | | | | | | | | | | | | | | | | | | | |
|_ | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
CROSSREFS
For the definition of "subparts" see A279387.
For the triangle of sums of subparts see A279388.
Cf. A317293 (number of vertices).
Cf. A060831 (number of regions or subparts).
Compare with A317109 (analog for the diagram that contains only parts).
First differs from A317109 at a(6).
Sequence in context: A173522 A049420 A232996 * A276221 A265284 A055507
KEYWORD
nonn,more
AUTHOR
Omar E. Pol, Jul 27 2018
STATUS
approved