The triangular numbers (and higher triangular numbers) can be generated using this recurrence relationship:

These will form Pascal’s triangle (if we shift the variables *n*->*n*+*d*-1 and *d*->*r*, we get the familiar C(*n*,*r*) indexing for the Pascal Triangle). The *d*=2 case gives the usual “flat” 2d triangular numbers, and other *d* values provide triangular numbers of different dimensions.

It turns out that recurrence relation can be generalized to generate a family of sequences and triangles. Consider this more general relation:

Doing some initial exploring reveals four interesting cases:

**The triangular numbers **

With all these additional parameters set to 1, we get our original relation, the familiar triangular numbers, and Pascal’s triangle.

**The k-polygonal numbers **

If we set the “zero dimension” to *k*-2, we end up with the *k*-polygonal numbers. The triangular numbers arise in the special case where *k*=3. Except in the *k*=3 case, the triangles that are generated are not symmetrical.

Below is the triangle generated by setting *k*=5.

**The symmetrically shifted k-polygonal numbers**

As far as I know, there is not a standard name for these. Each *k* value will generate a triangle that is symmetrical about its center and whose edge values are equal to *k*-2. For a given *k* value, if you enter sequences generated by particular values of *d*, you’ll find that some are well known. The codes in the diagrams correspond to the sequence ids from the Encyclopedia.

Here is the triangle generated by *k*=4:

And here is the triangle generated for *k*=5:

**The Eulerian numbers (Euler’s number triangle)**

This is a particularly nice way to generate the Eulerian numbers, which have a nice connection to the triangular numbers. There is a little inconsistency in the way the Eulerian numbers are indexed, however. For this formula to work, it should be altered slightly so that *d*>0. The resulting formula looks like this:

And the triangle looks like this:

It is surprising that so many interesting and well known sequences and triangles can be generated from such a simple formula, and that they can be interpreted as being part of a single family.

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*Credit for article given to dan.mackinnon*****