In calculating logarithms, square roots and value of we will begin by considering the Pascal triangle:

**Logarithms:**

To calculate the natural logarithm of consider the following triangle:

Multiply elementwise the Pascal triangle, matrix , with matrix so that we get:

Calculate the matrix inverse of matrix . Then the first column will be a sequence

with a property such that converges to the natural logarithm of , .

Example: For we get

and which is approximately equal to

**Square roots:**

To calculate the square root of we consider this triangle:

and this triangle:

Multiply elementwise the Pascal triangle with matrix and divide elementwise with matrix . We should then get:

we calculate the matrix inverse of matrix . Then the first column will be a sequence

with a property such that converges to the square root of , .

Example: For we get

and which is approximately equal to

**Value of **

Consider the following table defined by:

n>=k: if (n-k) modulo 4 = 1 or if (n-k) modulo 4 = 2 then -1 else 1

Multiply matrix elementwise with the Pascal triangle matrix . We then get:

Calculate the matrix inverse of so that we get:

The first column then has the property that converges to , .

Example:

and which is approximately equal to .

Keywords: Matrix inversion.

Very cool. I just created 3DListPlot of this in Mathematica, wrapped in a Manipulate, for visualization…just scanning the internet for interesting links to attach to my demo – for their (Mathworld.com) site.

Thanks,

Another number triangle with the same properties although somewhat slower convergence is the partial column sums of the Mahonian numbers defined by: