Polynomiluvut

Polynomiluvut, also known as polygonal numbers, are a class of figurate numbers that can be represented as a pattern of dots arranged in a regular polygon. These numbers arise from the geometric construction of polygons.

The simplest polygonal numbers are the triangular numbers, which represent dots arranged in an equilateral triangle. The nth triangular number is given by the formula T(n) = n(n+1)/2. For example, the first few triangular numbers are 1, 3, 6, 10, and 15.

Quadratic numbers, or square numbers, represent dots arranged in a square. The nth square number is given by the formula S(n) = n^2. The sequence begins with 1, 4, 9, 16, and 25.

Pentagonal numbers represent dots arranged in a regular pentagon. The formula for the nth pentagonal number is P(n) = n(3n-1)/2. The sequence starts with 1, 5, 12, 22, and 35.

Hexagonal numbers represent dots arranged in a regular hexagon. The formula for the nth hexagonal number is H(n) = n(2n-1). The sequence begins with 1, 6, 15, 28, and 45.

In general, the formula for the kth polygonal number with s sides is given by P(s, k) = k((s-2)k - (s-4))/2. This formula can be used to generate any polygonal number given the number of sides of the polygon and the position in the sequence.

Polygonal numbers have connections to various areas of mathematics, including number theory and combinatorics. They are a fundamental concept in the study of figurate numbers and their geometric interpretations.