A336305 Alternating row sums of triangle A211343.

1, 2, 2, 4, 3, 7, 4, 8, 7, 9, 6, 15, 7, 12, 13, 16, 9, 20, 10, 22, 16, 18, 12, 31, 16, 21, 20, 29, 15, 37, 16, 32, 24, 27, 25, 46, 19, 30, 28, 46, 21, 49, 22, 42, 40, 36, 24, 63, 29, 47, 36, 49, 27, 61, 36, 61, 40, 45, 30, 85, 31, 48, 53, 64, 42, 73, 34, 63, 48, 73, 36, 99, 37

Offset: 1

Author

Omar E. Pol, Oct 05 2020

Keywords

• nonn

Comments

On the infinite square grid the diagram of a(n) in the fourth quadrant is the same as the diagram of the symmetric representation of sigma(n), but taken only the part that is located in one of the octants (for example on the 7th octant), including totally the unit square cells that are on the main diagonal of the structure (see example).

The number of cells on the main diagonal of the diagram equals A067742(n).

The indices of the regions that have edges on the right border of the diagram give A071562.

a(n) = n if and only if n is a power of 2.

The diagram of a(n) is easily visible in the terraces of the n-th level (starting from the top) of the stepped pyramid described in A245092 (see Links section).

Note that some sequences as A000203, A067742, this sequence and many others appears to be more related to the double-staircases diagram of A196020 and to the horizontal faces of the pyramid, while many other sequences appears to be more related to the double-staircases diagram of A237593 and to the vertical faces of the pyramid. Both diagrams appears to be essentially the same, but they are not exactly equal.

Examples

			Illustration of initial terms:
   n    a(n)    _
   1     1     |_|_
   2     2     |_ _|_
   3     2     |_ _| |_
   4     4     |_ _ _| |_
   5     3     |_ _ _|  _|_
   6     7     |_ _ _ _|  _|_
   7     4     |_ _ _ _| |_ _|_
   8     8     |_ _ _ _ _|  _| |_
   9     7     |_ _ _ _ _| |     |_
  10     9     |_ _ _ _ _ _|  _ _| |_
  11     6     |_ _ _ _ _ _| |  _|  _|_
  12    15     |_ _ _ _ _ _ _| |_ _|  _|
  13     7     |_ _ _ _ _ _ _| |  _ _|
  14    12     |_ _ _ _ _ _ _ _| |
  15    13     |_ _ _ _ _ _ _ _| |
  16    16     |_ _ _ _ _ _ _ _ _|
...
		

Links

• Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Crossrefs

Cf. A000079, A000203, A067742, A071562, A196020, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A286001, A335616.

Formula

Conjecture: a(n) = (A000203(n) + A067742(n))/2.