Stjärnantalet

Stjärnantalet is a Swedish term that translates to "star number" or "stellar number." It refers to a specific type of figurate number, a concept in recreational mathematics. Figurate numbers represent geometric arrangements of discrete objects. A star number is formed by a pattern of dots arranged in a six-pointed star. The sequence of star numbers begins with 1, 13, 37, 73, 121, and so on. The formula for the n-th star number is given by $S_n = 6n(n-1) + 1$. This formula can also be expressed as $S_n = 3n^2 - 3n + 1$. These numbers appear in various mathematical contexts, including number theory and geometry. For instance, the first star number, 1, represents a single point. The second star number, 13, can be visualized as a central point surrounded by two layers of points forming a six-pointed star. The pattern continues with each successive star number being formed by adding a new layer of points to the previous star. The concept of figurate numbers, including star numbers, has a long history, with early investigations dating back to ancient Greek mathematicians.