New transforms

[Dean Hickerson]

Neil Sloane wrote:

> 2. Take the sum of 2 or more consecutive terms of a sequence.
...
> SeqFans are invited to supply further examples of all four transforms.

This isn't exactly what Neil meant, since I'm taking sums of exactly 2 terms.

Sequence A002623 contains the power series coefficients of 1/((1-x)^3*(1-x^2)),
which is also the number of nondegenerate triangles that can be made from
rods of length 1,2,3,4,...,n: 1, 3, 7, 13, 22, 34, 50, ...

Sums of 2 consecutive terms of that gives sequence A000292, the tetrahedral
numbers: 1, 4, 10, 20, 35, 56, 84, ...

Sums of 2 consecutive terms of that gives A000330, the square pyramidal
numbers: 1, 5, 14, 30, 55, 91, 140, ...

Sums of 2 consecutive terms of that gives A005900, the octahedral numbers:
1, 6, 19, 44, 85, 146, 231, ...

Sums of 2 consecutive terms of that gives A001845, the centered octahedral
numbers: 1, 7, 25, 63, 129, 231, 377, ...

Sums of 2 consecutive terms of that gives A008412, "Coordination sequence
for 4-dimensional cubic lattice (points on surface of 4-dimensional
cross-polytope)": 1, 8, 32, 88, 192, 360, 608, ...

Are there any longer chains of sequences in the database that are related
this way?

Dean Hickerson
dean at math.ucdavis.edu