A005226

A005226 is an integer sequence in the On-Line Encyclopedia of Integer Sequences (OEIS). It is defined as the sequence of central polygonal numbers, which are numbers that represent the number of dots in a pattern of nested polygons with a common center. Specifically, the nth term of the sequence, denoted as a(n), is given by the formula a(n) = n(n-1)/2 + 1.

The sequence begins with 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, and so on. These numbers can be visualized as arranging dots in layers around a central dot. The first layer has 1 dot (the center). The second layer adds 1 dot to form a line (2 dots total). The third layer adds 2 dots to form a triangle (4 dots total). The fourth layer adds 3 dots to form a square (7 dots total), and so forth. Each subsequent layer adds one more dot than the previous layer.

Central polygonal numbers have various applications in combinatorics and number theory. They are related to figurate numbers and can be found in patterns related to triangulations and partitions. The difference between consecutive terms in the sequence A005226 is simply n-1, where n is the term number. This means the sequence grows quadratically.