title: Dodecagonal number
text: # Dodecagonal number
  
  A dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for n is given by the formula
  
      $D_{n}=5n^{2}-4n$
  
  The first few dodecagonal numbers are:
  
      0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729, 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 ... (sequence A051624 in the OEIS)
  
  ## Properties
  
  * The dodecagonal number for n can be calculated by adding the square of n to four times the (n - 1)th pronic number, or to put it algebraically, $D_{n}=n^{2}+4(n^{2}-n)$.
  * Dodecagonal numbers consistently alternate parity, 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.
  