A294723 - OEIS

%I #86 Dec 31 2020 11:11:15

%S 1,4,7,11,16,20,27,31,38,45,53,57,66,70,78,89,100,104,115,119,130,142,

%T 150,154,167,176,184,196,211,215,230,234,249,261,269,280,297,301,309,

%U 321,338,342,359,363,379,398,406,410,429,440,459,471,487,491,510

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

%C 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 Robert Price, <a href="/A294723/b294723.txt">Table of n, a(n) for n = 0..5000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">An infinite stepped pyramid</a>

%H 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 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (first 16 levels)</a>

%F a(n) = A317109(n) - A237590(n) + 1 (Euler's formula). - _Omar E. Pol_, Jul 21 2018

%e Illustration of initial terms (n = 0..9):

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%e Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):

%e ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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%e |_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |

%e |_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |

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%e |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|

%e .

%Y Cf. A317109 (number of edges).

%Y Cf. A237590 (number of regions or parts).

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

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

%K nonn

%O 0,2

%A _Omar E. Pol_, Nov 07 2017

%E Terms a(30) and beyond from _Robert Price_, Jul 31 2018

%E Example extended for a(7)-a(9) and a(29) by _Omar E. Pol_, Jul 31 2018