A294723 a(n) is the total number of vertices after n-th stage in the diagram of the symmetries of sigma described in A236104, with a(0) = 1.

1, 4, 7, 11, 16, 20, 27, 31, 38, 45, 53, 57, 66, 70, 78, 89, 100, 104, 115, 119, 130, 142, 150, 154, 167, 176, 184, 196, 211, 215, 230, 234, 249, 261, 269, 280, 297, 301, 309, 321, 338, 342, 359, 363, 379, 398, 406, 410, 429, 440, 459, 471, 487, 491, 510

Offset: 0

Author

Omar E. Pol, Nov 07 2017

Keywords

• nonn

Comments

a(n) is also the total number of "hinges" in the "mechanism" where every row of the two-dimensional diagram of the isosceles triangle with n rows described in A237593 is folded in a 90-degree zig-zag, appearing the structure of the stepped pyramid with n levels described in A245092. Note that the diagram described in A236104 is also the top view of the mentioned pyramid. The area of the terraces in the n-th level of the pyramid, starting from the top, equals sigma(n) = A000203(n).

For the construction of the two-dimensional diagram using Dyck paths and for more information about the pyramid see A237593 and A262626.

Note that every line segment of the Dyck paths of the diagram is related to partitions into consecutive parts (see A237591). - Omar E. Pol, Feb 23 2018

Examples

			Illustration of initial terms (n = 0..9):
.                                                           _ _ _ _
.                                             _ _ _        |_ _ _  |_
.                                 _ _ _      |_ _ _|       |_ _ _|   |_
.                       _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.               _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.         _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    .   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.    1    4      7        11         16           20             27
.
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _|   |_ _      |_ _ _|   |_ _  |     |_ _ _|   |_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           31                  38                     45
.
.
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
		

Links

• Robert Price, Table of n, a(n) for n = 0..5000
• Omar E. Pol, An infinite stepped pyramid
• Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
• Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Crossrefs

Cf. A317109 (number of edges).

Cf. A237590 (number of regions or parts).

Compare with A317293 (analog for the diagram that contains subparts).

Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A294847, A296508.

Formula

a(n) = A317109(n) - A237590(n) + 1 (Euler's formula). - Omar E. Pol, Jul 21 2018

Extensions

Terms a(30) and beyond from Robert Price, Jul 31 2018

Example extended for a(7)-a(9) and a(29) by Omar E. Pol, Jul 31 2018