Here we will look at the 3sided Pascalloid or triangular stepped pyramid of numbers. The combinatorics equation achieves the numbers with precision and at less computational cost. The recursive number algorithm that also gives rise to our triangular tiling patterns is given by this shorthand, it is ideally tripolar:
Triangular Number Summation output to N=12
Starting again with the number 'one', a triangular computation algorithm also known as Nugent's Pyramid leads to the following sheets of numbers. They line up to form a 3sided pyramid. The base of this pyramid is the sheet of numbers for (A+B+C)^{N}. This relates directly to Trinomial Theorem. The Combinatorics equation is as follows:
Download
XL file:
Triangle
Computation Simplified.xls
Note that each of the three variables forming the terms is equal to unity, preserving just the coefficients.
N = 0



N = 1



N = 2



N = 3



N = 4



N = 5



N = 6



N = 7



N = 8



N = 9



N = 10



N = 11



N = 12



N = 13



N = 14



N = 15



N = 16



N = 17



N = 18



N = 19



N = 20



N = 21



N = 22



N = 23



N = 24



N = 25



N = 26



N = 27



N = 28



N = 29



N = 30



N = 31



N = 32



N = 33



N = 34



It is of note that half the time 2/3 of these numbers are redundant information, and half the time [total numbers displayed at chosen at any even valued N * 2/3  1] of these numbers are redundant. If the correct terms are included (they are derivable) then I suggest it is like a Yantra for balance. However the building blocks of polynomial terms are relatively simple.