A317293 a(n) is the total number of vertices 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) = 1.

1, 4, 7, 11, 16, 20, 28, 32, 39, 46, 54, 58, 72, 76, 84, 96, 107, 111, 126, 130, 144, 156, 164, 168, 190, 199, 207, 219, 235, 239

Offset: 0

Author

Omar E. Pol, Jul 27 2018

Keywords

• nonn
• more

Comments

Note that in the diagram the number of regions or subparts equals A060831, the partial sums of A001227, n >= 1.

Examples

			Illustration of initial terms (n = 0..9):
.                                                           _ _ _ _
.                                             _ _ _        |_ _ _  |_
.                                 _ _ _      |_ _ _|       |_ _ _| |_|_
.                       _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.               _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.         _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    .   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.    1    4      7        11         16           20             28
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _| |_|_ _      |_ _ _| |_|_ _  |     |_ _ _| |_|_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           32                  39                     46
.
.
Illustration of the two-dimensional diagram after 29 stages (contains 239 vertices, 300 edges and 62 regions or subparts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | | |_ _  |_ _| |_|_
|_ _ _ _ _ _ _ _ _ _| | |_ _  | |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _| |_  |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_  |_  |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_  |_ _  | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| | |_ _| |_  | |_ _  | | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_  |_|_  | | |_|_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_  | | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _| |_|_  | |_|_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  | |_  |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_  |_  | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_  |_|_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _| |_|_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
		

Crossrefs

For the definition of "subparts" see A279387.

For the triangle of sums of subparts see A279388.

Cf. A317292 (number of edges).

Cf. A060831 (number of regions or subparts).

Compare with A294723 (analog for the diagram that contains only parts).

First differs from A294723 at a(6).

Cf. A000203, A196020, A235791, A237048, A237590, A237591, A237270, A237271, A237593, A245092, A244050, A262626, A280850, A280851, A280940, A285901, A296508, A317109.

Formula

a(n) = A317292(n) - A060831(n) + 1 (Euler's formula).