Triangular array

In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index.

Notable particular examples include these: Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called generalized Pascal triangles; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.

[13] In general, a triangular array is used to store any table indexed by two natural numbers where j ≤ i. Storing a triangular array in a computer requires a mapping from the two-dimensional coordinates (i, j) to a linear memory address.

If two triangular arrays of equal size are to be stored (such as in LU decomposition), they can be combined into a standard rectangular array.

Just like a rectangular array, one multiplication is required to find the start of the row, but this multiplication is of two variables (i*(i+1)/2), so some optimizations such as using a sequence of shifts and adds are not available.