A300699 -id:A300699 - cp's OEIS frontend

A300695 Irregular triangle read by rows: T(n, k) = number of vertices with rank k in cocoon concertina n-cube.

1, 1, 1, 1, 3, 3, 1, 1, 6, 13, 6, 13, 6, 1

Offset: 0

Tilman Piesk, Mar 13 2018

Although the cocoon concertina n-cube has no ranks for n>2, its inner vertices can be forced on the rank layers of the convex solid.
Sum of row n is the number of vertices of a cocoon concertina n-cube, i.e., A000696(n).
The rows are palindromic. Their lengths are the central polygonal numbers A000124 = 1, 2, 4, 7, 11, 16, 22, ... That means after row 0 rows of even and odd length follow each other in pairs.
A300699 is a triangle of the same shape that shows the number of ranks in convex concertina hypercubes.

			First rows of the triangle:
    k   0    1    2    3    4    5    6
  n
  0     1
  1     1    1
  2     1    3    3    1
  3     1    6   13    6   13    6    1

Tilman Piesk, ranks 1 / 5, 2 / 4 and 3 for n=3
Tilman Piesk, Python code used to generate the sequence (currently unfinished, does not find all ranks for n>3)

Cf. A000696, A300699, A000124.

cp's OEIS Frontend

A300695 Irregular triangle read by rows: T(n, k) = number of vertices with rank k in cocoon concertina n-cube.

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