A279029 Numbers k with the property that the smallest and the largest Dyck path of the symmetric representation of sigma(k) do not share line segments.

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 136, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 256

Offset

1,2

Comments

Numbers k such that the symmetric representation of sigma(k) is formed by only one part, or that it's formed by only two parts and they meet at the center.

Numbers k whose total length of all line segments of the symmetric representation of sigma(k) is equal to 4*k (cf. A348705). For the positive integers k that are not in this sequence the mentioned total length is < 4*k. - Omar E. Pol, Nov 02 2021

From Hartmut F. W. Hoft, Jan 25 2025: (Start)

The following three statements are equivalent for numbers k >= 1:

(1) The symmetric representation of sigma(k) is formed of 2 parts that meet at the diagonal.

(2) A249223(k, A003056(k)) = 0 is the only 0 in row k of the triangle, and A237591(k, A003056(k)) = 1.

(3) Row k of the triangle in A341969 contains a single 0 at the center position.

The following two statements are equivalent for numbers k >= 1:

(1) The symmetric representation of sigma(k) consists of a single part.

(2) Row k of the triangle in A249223 contains no 0. (End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = A323648(n-1) + 1, for n >= 2. - Hartmut F. W. Hoft, Jan 25 2025

Example

1, 2, 3, 4, 6, 8, 10, 12 and 16 are in the sequence because the smallest and the largest Dyck path of their symmetric representation of sigma do not share line segments, as shown below.
llustration of initial terms:
n
.      _ _ _ _   _   _   _   _       _
1     |_| | | | | | | | | | | |     | |
2     |_ _|_| | | | | | | | | |     | |
3     |_ _|  _|_| | | | | | | |     | |
4     |_ _ _|    _|_| | | | | |     | |
       _ _ _|  _|  _ _|_| | | |     | |
6     |_ _ _ _|  _| |  _ _|_| |     | |
       _ _ _ _| |_ _|_|    _ _|     | |
8     |_ _ _ _ _|  _|     |    _ _ _| |
       _ _ _ _ _| |      _|   |  _ _ _|
10    |_ _ _ _ _ _|  _ _|    _| |
       _ _ _ _ _ _| |      _|  _|
12    |_ _ _ _ _ _ _|  _ _|  _|
                      |  _ _|
                      | |
       _ _ _ _ _ _ _ _| |
16    |_ _ _ _ _ _ _ _ _|
...

Mathematica

(* Function path[] is defined in A237270 *)
a279029Q[n_] := Length[Select[Transpose[{Take[path[n], {2, -2}], path[n-1]}], #[[1]]==#[[2]]&]]<=1
a279029[n_] := Select[Range[n], a279029Q]
a279029[256]
(* Alternate, faster function based on A249223 *)
a003056[n_] := Floor[(Sqrt[8n+1]-1)/2]
t249223[n_] :=FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, a003056[n]]]
a262259Q[n_] := Position[t249223[n], 0]=={{a003056[n]}}&&Last[t237591[n]]==1
a174973Q[n_] := !MemberQ[t249223[n], 0]
a279029[n_] := Select[Range[n], a262259Q[#]||a174973Q[#]&]
a279029[256] (* Hartmut F. W. Hoft, Jan 25 2025 *)

Prog

(PARI) is_A279029 = A379968; \\ Antti Karttunen, Jan 12 2025

Crossrefs

UNION of A174973 and A262259.

Positions of 0's in A279228.

Complement is A279244.

Cf. A000203, A003056, A008586, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A249223, A262259, A323648, A348705, A379968 (characteristic function).

Sequence in context: A359754 A330899 A325796 * A316886 A309943 A002093

Adjacent sequences: A279026 A279027 A279028 * A279030 A279031 A279032

Keyword

nonn

Author

Omar E. Pol, Dec 08 2016

Status

approved