A276112 Numbers with precipice 1: descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to 1.

1, 3, 5, 7, 8, 11, 14, 15, 17, 19, 23, 24, 27, 29, 31, 34, 35, 39, 41, 44, 47, 48, 49, 53, 55, 59, 62, 63, 65, 69, 71, 76, 79, 80, 83, 87, 89, 90, 95, 97, 98, 99, 103, 107, 109, 111, 116, 119, 120, 125, 127, 129, 131, 134, 139, 142, 143, 149, 152, 153, 155, 159

Offset

1,2

Comments

The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.

The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k).

The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.

For more information about the precipices see A277437, A280223 and A280295.

From Hartmut F. W. Hoft, Feb 02 2022: (Start)

Also partial sums of A280919.

a(n) is also the largest number of a Dyck path that crosses the diagonal at point A282131(n) which also is the rightmost number in each nonzero row of the irregular triangle in A279385. (End)

Links

Table of n, a(n) for n=1..62.

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

Formula

a(n) = A071562(n+1) - 1.

a(n) = Sum_{i=1..n} A280919(i), n >= 1. - Hartmut F. W. Hoft, Feb 02 2022

Example

From Hartmut F. W. Hoft, Feb 02 2022: (Start)
      n: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 index.
A282131: 1  2  3  5  6  7  9 11 12 13 15 17 18 20 position on diagonal.
A276112: 1  3  5  7  8 11 14 15 17 19 23 24 27 29 max index of Dyck path.
A280919: 1  2  2  2  1  3  3  1  2  2  4  1  3  2 paths at diag position.
(End)

Mathematica

(* last computed value of a280919[ ] is dropped to avoid a potential undercount of crossings *)
a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]
a280919[n_] := Most[Map[Length, Split[Map[a240542, Range[n]]]]]
A276112[160] (* Hartmut F. W. Hoft, Feb 02 2022 *)

Crossrefs

Column 1 of A277437.

Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223, A280295.

Cf. A280919, A282131.

Sequence in context: A024969 A296233 A371185 * A228075 A309405 A354300

Adjacent sequences: A276109 A276110 A276111 * A276113 A276114 A276115

Keyword

nonn

Author

Omar E. Pol, Jan 02 2017

Status

approved