Figurate Number - Triangular Numbers

Triangular Numbers

The triangular numbers for n = 1, 2, 3, ... are the result of the juxtaposition of the linear numbers (linear gnomons) for n = 1, 2, 3, ...:

These are the binomial coefficients . This is the case r=2 of the fact that the rth diagonal of Pascal's triangle for consists of the figurate numbers for the r-dimensional analogs of triangles (r-dimensional simplices).

The simplicial polytopic numbers for r = 1, 2, 3, 4, ... are:

• (linear numbers),
• (triangular numbers),
• (tetrahedral numbers),
• (pentachoron numbers, pentatopic numbers, 4-simplex numbers),

• (r-topic numbers, r-simplex numbers).

The terms square number and cubic number derive from their geometric representation as a square or cube. The difference of two positive triangular numbers is a trapezoidal number.