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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.

%I A294723 #86 Dec 31 2020 11:11:15
%S A294723 1,4,7,11,16,20,27,31,38,45,53,57,66,70,78,89,100,104,115,119,130,142,
%T A294723 150,154,167,176,184,196,211,215,230,234,249,261,269,280,297,301,309,
%U A294723 321,338,342,359,363,379,398,406,410,429,440,459,471,487,491,510
%N 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.
%C A294723 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).
%C A294723 For the construction of the two-dimensional diagram using Dyck paths and for more information about the pyramid see A237593 and A262626.
%C A294723 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
%H A294723 Robert Price, <a href="/A294723/b294723.txt">Table of n, a(n) for n = 0..5000</a>
%H A294723 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">An infinite stepped pyramid</a>
%H A294723 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)</a>
%H A294723 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (first 16 levels)</a>
%F A294723 a(n) = A317109(n) - A237590(n) + 1 (Euler's formula). - _Omar E. Pol_, Jul 21 2018
%e A294723 Illustration of initial terms (n = 0..9):
%e A294723 .                                                           _ _ _ _
%e A294723 .                                             _ _ _        |_ _ _  |_
%e A294723 .                                 _ _ _      |_ _ _|       |_ _ _|   |_
%e A294723 .                       _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
%e A294723 .               _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
%e A294723 .         _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
%e A294723 .    .   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
%e A294723 .
%e A294723 .    1    4      7        11         16           20             27
%e A294723 .
%e A294723 .
%e A294723 .                                               _ _ _ _ _
%e A294723 .                         _ _ _ _ _            |_ _ _ _ _|
%e A294723 .     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
%e A294723 .    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
%e A294723 .    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
%e A294723 .    |_ _ _|   |_ _      |_ _ _|   |_ _  |     |_ _ _|   |_ _  | |
%e A294723 .    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
%e A294723 .    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
%e A294723 .    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
%e A294723 .    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
%e A294723 .
%e A294723 .           31                  38                     45
%e A294723 .
%e A294723 .
%e A294723 Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
%e A294723 ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A294723 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A294723 |_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
%e A294723 |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e A294723 |_ _ _ _ _ _ _ _ _ _ _ _ _  | |
%e A294723 |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e A294723 |_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
%e A294723 |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
%e A294723 |_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
%e A294723 |_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
%e A294723 |_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
%e A294723 |_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
%e A294723 |_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
%e A294723 |_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
%e A294723 |_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
%e A294723 |_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
%e A294723 |_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
%e A294723 |_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
%e A294723 |_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
%e A294723 |_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
%e A294723 |_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
%e A294723 |_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
%e A294723 |_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
%e A294723 |_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
%e A294723 |_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
%e A294723 |_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
%e A294723 |_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
%e A294723 |_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
%e A294723 |_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
%e A294723 |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
%e A294723 .
%Y A294723 Cf. A317109 (number of edges).
%Y A294723 Cf. A237590 (number of regions or parts).
%Y A294723 Compare with A317293 (analog for the diagram that contains subparts).
%Y A294723 Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A294847, A296508.
%K A294723 nonn
%O A294723 0,2
%A A294723 _Omar E. Pol_, Nov 07 2017
%E A294723 Terms a(30) and beyond from _Robert Price_, Jul 31 2018
%E A294723 Example extended for a(7)-a(9) and a(29) by _Omar E. Pol_, Jul 31 2018