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A364414 Numbers k with the property that the second part of the symmetric representation of sigma(k) is an octagon of width 1 and one of its vertices is also the central vertex of the first valley of the largest Dyck path of the diagram.

%I A364414 #24 Sep 03 2023 10:53:24
%S A364414 21,27,33,39,51,57,63,69,81,87,93,99,111,117,123,129,141,147,153,159,
%T A364414 171,177,183,189,201,207,213,219,231,237,243,249,261,267,273,279,291,
%U A364414 297,303,309,321,327,333,339,351,357
%N A364414 Numbers k with the property that the second part of the symmetric representation of sigma(k) is an octagon of width 1 and one of its vertices is also the central vertex of the first valley of the largest Dyck path of the diagram.
%C A364414 Conjecture 1: These are the numbers > 9 that are congruent to {3, 9, 21, 27} mod 30.
%C A364414 Conjecture 2: These are the terms > 9 of A016945 except the terms ending in 5.
%C A364414 Conjecture 3: The polygon mentioned in the definition is an "S"-shaped concave octagon.
%C A364414 Conjecture 4: Every term of this sequence has as nearest neighbor a term of A091999.
%C A364414 Conjecture 5: The terms of A091999 greater than 2 are the numbers k with the property that the first part of the symmetric representation of sigma(k) is an octagon.
%C A364414 Conjecture 6: The octagon mentioned in the definition shares at least an edge with the octagon mentioned in conjecture 5.
%C A364414 Also the row numbers of the triangle A364639 where the rows start with [0, 0, 1, 0, -1]. - _Omar E. Pol_, Aug 23 2023
%e A364414 The symmetric representation of sigma(21) in the first quadrant looks like this:
%e A364414    _ _ _ _ _ _ _ _ _ _ _
%e A364414   |_ _ _ _ _ _ _ _ _ _ _|
%e A364414                         |
%e A364414                         |
%e A364414                         |_ _ _
%e A364414                         |_ _  |_
%e A364414                             |_ _|_
%e A364414                                 | |_
%e A364414                                 |_  |
%e A364414                                   | |
%e A364414                                   |_|_ _ _ _
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           | |
%e A364414                                           |_|
%e A364414 .
%e A364414 Its second part is an octagon of width 1 and one of its vertices is also the central vertex of the first valley of the largest Dyck path of the structure, so 21 is in the sequence.
%e A364414 Note that 10 is not in the sequence because the second part of the symmetric representation of sigma(10) is an octagon of width 1 in accordance with the definition but none of its vertices is the central vertex of the first valley of the largest Dyck path of the diagram.
%Y A364414 Subsequence of A016945.
%Y A364414 Cf. A000203, A001068, A045572, A091999, A196020, A235791, A236104, A237270 (parts), A237271, A237591, A237593, A244145, A245092, A249223, A249351
%Y A364414 (widths), A262626, A362866, A364639, A365081.
%K A364414 nonn
%O A364414 1,1
%A A364414 _Omar E. Pol_, Jul 23 2023