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.
Read more about this topic: Figurate Number
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