cp's OEIS Frontend

A279228 Number of unit steps that are shared by the smallest and largest Dyck path of the symmetric representation of sigma(n).

%I A279228 #26 Dec 18 2021 22:20:48
%S A279228 0,0,0,0,2,0,4,0,2,0,8,0,10,2,4,0,14,0,16,0,6,6,20,0,16,8,10,0,26,0,
%T A279228 28,0
%N A279228 Number of unit steps that are shared by the smallest and largest Dyck path of the symmetric representation of sigma(n).
%C A279228 a(n) is also the number of unit steps that are shared by the largest Dyck path of the symmetric representation of sigma(n) and the largest Dyck path of the symmetric representation of sigma(n-1), in a quadrant of the square grid.
%C A279228 For more information about the Dyck paths of the symmetric representation of sigma(n) see A237593.
%F A279228 a(n) = 2*n - A244363(n) = 2*(n - A244361(n)).
%F A279228 a(n) = A008586(n) - A348705(n). - _Omar E. Pol_, Dec 13 2021
%e A279228 Illustration of initial terms (n = 1..12) using the spiral described in A239660:
%e A279228 .               _ _ _ _ _ _
%e A279228 .              |  _ _ _ _ _|_ _ _ _ _
%e A279228 .         0   _| |         |_ _ _ _ _|
%e A279228 .           _|_ _|                   |_ _ 2
%e A279228 .       _ _| |      _ _ _ _          |_  |
%e A279228 .      |  _ _|  0 _|  _ _ _|_ _ _      |_|_ _
%e A279228 .      | |      _|   |     |_ _ _|  2      | |
%e A279228 .      | |     |  _ _|           |_ _      | |
%e A279228 .      | |     | |    0 _ _        | |     | |
%e A279228 .      | |     | |     |  _|_ 0    | |     | |
%e A279228 .     _|_|    _|_|    _|_| |_|    _|_|    _|_|    _
%e A279228 .    | |     | |     | |         | |     | |     | |
%e A279228 .    | |     | |     |_|_ _     _| |     | |     | |
%e A279228 .    | |     | |      0|_ _|_ _|  _|     | |     | |
%e A279228 .    | |     |_|_          |_ _ _|0   _ _| |     | |
%e A279228 .    | |         |_                 _|  _ _|     | |
%e A279228 .    |_|_ _     4  |_ _ _ _        |  _|    _ _ _| |
%e A279228 .          |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|
%e A279228 .            |_            |_ _ _ _ _|  _|     |
%e A279228 .         8    |                       |      _|
%e A279228 .              |_ _ _ _ _ _            |  _ _|
%e A279228 .              |_ _ _ _ _ _|_ _ _ _ _ _| |      0
%e A279228 .                          |_ _ _ _ _ _ _|
%e A279228 .
%e A279228 .
%e A279228 For an illustration of the following examples see the last lap of the above spiral starting in the first quadrant.
%e A279228 For n = 9 the Dyck paths of the symmetric representation of sigma(9) share 2 unit steps, so a(9) = 2.
%e A279228 For n = 10 the Dyck paths of the symmetric representation of sigma(10) meet at the center, but they do not share unit steps, so a(10) = 0.
%e A279228 For n = 11 the Dyck paths of the symmetric representation of sigma(11) share 8 unit steps, so a(11) = 8.
%e A279228 For n = 12 the Dyck paths of the symmetric representation of sigma(12) do not share unit steps, so a(12) = 0.
%e A279228 Note that we can find the spiral on the terraces of the stepped pyramid described in A244050.
%Y A279228 Cf. A279029 gives the indices of the zero values.
%Y A279228 Cf. A279244 gives the indices of the positive values.
%Y A279228 Cf. A000203, A005843, A008586, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A244050, A244361, A244363, A245092, A262626, A348705.
%K A279228 nonn,more
%O A279228 1,5
%A A279228 _Omar E. Pol_, Dec 08 2016