Metallic Ratios and Pascal’s Triangle : Triads of Metallic Means in the Pascal’s Triangle

This paper introduces the close correspondence between Pascal’s Triangle and the recently published mathematical formulae those provide the precise relations between different Metallic Ratios. The precise correlations between various Metallic Means can be substantiated with Pascal’s Triangle, as described herein.


Introduction
The renowned Pascal's Triangle, having the binomial coefficients in its rows and the Fibonacci Numbers in its shallow diagonals, is much celebrated in algebra, probability theory, and combinatorics.
However, more importantly, now it has been observed that this intriguing Triangular array of numbers also highlights the precise mathematical relations between different Metallic Ratios.
The Mathematical Formulae those provide the precise relations between different Metallic Means have been recently published in the works mentioned in References [1] and [2].
It is also observed that the precise correlations between various Metallic Means given by those formulae, can also be substantiated with Pascal's Triangle.
The prime objective of this work is to bring together this Pascal's Triangle and a recently published formula that provides the accurate mathematical relations between different Metallic Means.
As a brief introduction, each Metallic Mean n is the root of the simple Quadratic Equation X 2 -nX -1 = 0, where n is any positive natural number.

MATHEMATICAL CORRELATIONS AMONG DIFFERENT METALLIC RATIOS :
If K, m and n are three positive integers such that n is the smallest of the three integers and Noticeably, n=6 forms such multiple triads: : Shaded Triads have been exemplified above.
And, just like n=6 exemplified above, every integer forms such multiple triads: It may be noticed from above Here, let us consider the close correspondence of the abovementioned Formula and TRIADS with the Pascal's Triangle.

Metallic Ratios and the Pascal's Triangle :
The abovementioned Formula and the TRIADS can be substantiated with the Pascal's Triangle, as follows.
Consider the integers n and m on the "Natural Numbers Diagonal" of the Pascal's Triangle, the precise value of k can be derived from the "Triangular Numbers Diagonal" of the Pascal's Triangle.  For instance, if m-n=2, then the couple of integers, say x and y, on the Triangular Numbers Diagonal, adjacent to n and m respectively, gives value of k : If n is ( 1 ) and m is ( +2 1 ); then x is ( +1 2 ) and y is ( +2 2 ) And here k = ( + )+ ; as illustrated below in Figure 3. i.e. n is ( 1 ) and m is ( +4 1 ); In this case, the integers w, x, y and z on the Triangular Numbers Diagonal, lying between n and m, with Row numbers from (n+1) to (m) i.e. Triangular Numbers from ( +1 2 ) to ( +4 2 ) give the value of k as : ; as shown below in Figure 4.    [12]. Further, all mathematical formulae those provide the precise relations between different Metallic Means are described in the work mentioned in Reference [1].

Conclusion:
This paper introduced the close correspondence between Pascal's Triangle and Metallic Means. The TRIADS of Metallic Means are produced by an imperical formula that provides the precise relations between different Metallic Ratios. And, such TRIADS of Metallic Means can also be substantiated with Pascal's Triangle, as described in this paper.