Generate a formula for an Octagonal figurative number

Question

 Generate a formula for the nth Octagonal number by relating to one or more figurate numbers with fewer than seven sides(by relating to hexagonal, pentagonal, square, oblong or triangular numbers).

Answer 1

The formulas for figurate numbers up to 6 are:

 

Triangular: (1/2)n² + (1/2)n

Square: n²

 

Pentagonal: (3/2)n² - (1/2)n

Hexagonal: 2n² - n

 

Notice that, with each step up in number, we're adding (1/2)n² and subtracting (1/2)n. That means that mathematically we'll have:

 

Heptagonal: 2n² - n + (1/2)n² - (1/2)n = (5/2)n² - (3/2)n

Octagonal: (5/2)n² - (3/2)n + (1/2)n² - (1/2)n = 3n² - 2n

 

That's one way to relate them.