A dodecagonal number is a

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polyg ...

that represents a

dodecagon In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational s ...

. The dodecagonal number for ''n'' is given by the formula :

D_=5n^2 - 4n

The first few dodecagonal numbers are: : 0, 1, 12, 33, 64, 105,

156 Year 156 (Roman numerals, CLVI) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Silvanus and Augurinus (or, less frequently, year 90 ...

, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548,

1729 Events January–March * January 8 – Frederick, the eldest son of King George II of Great Britain is made Prince of Wales at the age of 21, a few months after he comes to Britain for the first time after growing up in Hano ...

, 1920, 2121, 2332, 2553, 2784, 3025, 3276, 3537, 3808, 4089, 4380, 4681, 4992, 5313, 5644, 5985, 6336, 6697, 7068, 7449, 7840, 8241, 8652, 9073, 9504, 9945 ...

Properties

*The dodecagonal number for ''n'' can be calculated by adding the square of ''n'' to four times the (''n'' - 1)th

pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...

, or to put it algebraically,

D_n = n^2 + 4(n^2 - n)

. *Dodecagonal numbers consistently alternate

parity Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...

, and in base 10, their units place digits follow the pattern 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. *By the Fermat polygonal number theorem, every number is the sum of at most 12 dodecagonal numbers. *

D_n

is the sum of the first n natural numbers congruent to 1 mod 10. *

D_n

is the sum of all odd numbers from 4n+1 to 6n+1.

Properties

See also