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A317109
a(n) is the total number of edges after n-th stage in the diagram of the symmetries of sigma described in A236104, with a(0) = 0.
7
0, 4, 8, 14, 20, 26, 34, 40, 48, 58, 68, 74, 84, 90, 100, 114, 126, 132, 144, 150, 162, 178, 188, 194, 208, 220, 230, 246, 262, 268, 284, 290, 306, 322, 332, 346, 364, 370, 380, 396, 414, 420, 438, 444, 462, 484, 494, 500, 520, 534, 556, 572, 590, 596, 616, 636
OFFSET
0,2
COMMENTS
All terms are even numbers.
Note that the two-dimensional diagram is also the top view of the stepped pyramid with n levels described in A245092.
For the construction of the two-dimensional diagram using Dyck paths and for more information about the pyramid see A237593.
LINKS
FORMULA
a(n) = A294723(n) + A237590(n) - 1 (Euler's formula).
EXAMPLE
Illustration of initial terms (n = 1..9):
. _ _ _ _
. _ _ _ |_ _ _ |_
. _ _ _ |_ _ _| |_ _ _| |_
. _ _ |_ _ |_ |_ _ |_ _ |_ _ |_ _ |
. _ _ |_ _|_ |_ _|_ | |_ _|_ | | |_ _|_ | | |
. _ |_ | |_ | | |_ | | | |_ | | | | |_ | | | | |
. |_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 4 8 14 20 26 34
.
. _ _ _ _ _
. _ _ _ _ _ |_ _ _ _ _|
. _ _ _ _ |_ _ _ _ | |_ _ _ _ |_ _
. |_ _ _ _| |_ _ _ _| |_ |_ _ _ _| |_ |
. |_ _ _ |_ |_ _ _ |_ |_ _ |_ _ _ |_ |_|_ _
. |_ _ _| |_ _ |_ _ _| |_ _ | |_ _ _| |_ _ | |
. |_ _ |_ _ | | |_ _ |_ _ | | | |_ _ |_ _ | | | |
. |_ _|_ | | | | |_ _|_ | | | | | |_ _|_ | | | | | |
. |_ | | | | | | |_ | | | | | | | |_ | | | | | | | |
. |_|_|_|_|_|_|_| |_|_|_|_|_|_|_|_| |_|_|_|_|_|_|_|_|_|
.
. 40 48 58
.
.
Illustration of the diagram after 29 stages (contain 268 edges, 215 vertices and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _ | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _ | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _ |
|_ _ _ _ _ _ _ _ _ _ _ | | |_ _ | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_ |_
|_ _ _ _ _ _ _ _ _ _ | | |_ _| |_
|_ _ _ _ _ _ _ _ _ _| | |_ _ |_ |_ _ |_ _
|_ _ _ _ _ _ _ _ _ | |_ _ _| |_ | |_ _ |
|_ _ _ _ _ _ _ _ _| | |_ _ |_ |_|_ _ | |
|_ _ _ _ _ _ _ _ | |_ _ |_ _|_ | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| | | | |_ _ | |_|_ _ _ _ _ | |
|_ _ _ _ _ _ _ | |_ _ |_ |_ | | |_ _ _ _ _ | | | |
|_ _ _ _ _ _ _| |_ _ |_ |_ _ | | |_ _ _ _ _ | | | | | |
|_ _ _ _ _ _ | |_ |_ |_ | |_|_ _ _ _ | | | | | | | |
|_ _ _ _ _ _| |_ _| |_ | |_ _ _ _ | | | | | | | | | |
|_ _ _ _ _ | |_ _ | |_ _ _ _ | | | | | | | | | | | |
|_ _ _ _ _| |_ | |_|_ _ _ | | | | | | | | | | | | | |
|_ _ _ _ |_ _|_ |_ _ _ | | | | | | | | | | | | | | | |
|_ _ _ _| |_ | |_ _ _ | | | | | | | | | | | | | | | | | |
|_ _ _ |_ |_|_ _ | | | | | | | | | | | | | | | | | | | |
|_ _ _| |_ _ | | | | | | | | | | | | | | | | | | | | | |
|_ _ |_ _ | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_ | | | | | | | | | | | | | | | | | | | | | | | | | |
|_ | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
CROSSREFS
Cf. A294723 (number of vertices).
Cf. A237590 (number of regions or parts).
Compare with A317292 (analog for the diagram that contains subparts).
Sequence in context: A312685 A176949 A274968 * A173522 A049420 A232996
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jul 21 2018
EXTENSIONS
More terms and b-file from Robert Price, Jul 31 2018
STATUS
approved