Triangular numbers

Triangular numbers (of dimension 2) are defined by T(n) = T(n;2) = n(n+1)/2 = Binomial(n+1,2).

More generally, triangular numbers of dimension 1 are T(n;1) = n = Binomial(n,1) and triangular numbers of dimension d+1 are T(n;d+1) = T(1;d) + ... + T(n;d) = Binomial(n+d-1,d).

TriangularNumber (n :: RngIntElt, d :: RngIntElt)

-> RngIntElt

The nth triangular number in dimension d. Equal to Binomial(n+d-1,d).

TriangularNumber (n :: RngIntElt)

-> RngIntElt

The nth triangular number. Equal to Binomial(n+1,2) = n*(n+1)/2.

LinearToTriangularIndex (i :: RngIntElt, d :: RngIntElt)

-> []

Converts from linear to triangular indexing in dimension d. That is, returns the ith array [i1,i2,...,id] lexicographically with i1 ge i2 ge ... ge id ge 1.

LinearToTriangularIndex (i :: RngIntElt)

-> []

Converts from linear to triangular indexing. That is, returns the ith array [i1,i2] lexicographically with i1 ge i2 ge 1.

TriangularToLinearIndex (idx :: [RngIntElt], d :: RngIntElt)

-> RngIntElt

Converts from triangular indexing in dimension d to linear.

TriangularToLinearIndex (idx :: [RngIntElt])

-> RngIntElt

Converts from triangular indexing to linear.