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.

21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99, 111, 117, 123, 129, 141, 147, 153, 159, 171, 177, 183, 189, 201, 207, 213, 219, 231, 237, 243, 249, 261, 267, 273, 279, 291, 297, 303, 309, 321, 327, 333, 339, 351, 357

Offset: 1

Author

Omar E. Pol, Jul 23 2023

Keywords

• nonn

Comments

Conjecture 1: These are the numbers > 9 that are congruent to {3, 9, 21, 27} mod 30.

Conjecture 2: These are the terms > 9 of A016945 except the terms ending in 5.

Conjecture 3: The polygon mentioned in the definition is an "S"-shaped concave octagon.

Conjecture 4: Every term of this sequence has as nearest neighbor a term of A091999.

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.

Conjecture 6: The octagon mentioned in the definition shares at least an edge with the octagon mentioned in conjecture 5.

Also the row numbers of the triangle A364639 where the rows start with [0, 0, 1, 0, -1]. - Omar E. Pol, Aug 23 2023

Examples

			The symmetric representation of sigma(21) in the first quadrant looks like this:
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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.
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.
		

Crossrefs

Subsequence of A016945.

Cf. A000203, A001068, A045572, A091999, A196020, A235791, A236104, A237270 (parts), A237271, A237591, A237593, A244145, A245092, A249223, A249351

(widths), A262626, A362866, A364639, A365081.