In mathematics, a dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for n is given by the formula
D n = 5 n 2 − 4 n {\displaystyle 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, ... (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 ) {\displaystyle 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 {\displaystyle D_{n}} is the sum of the first n natural numbers congruent to 1 mod 10.
D n + 1 {\displaystyle D_{n+1}} is the sum of all odd numbers from 4n+1 to 6n+1.
Sum of reciprocals A formula for the sum of the reciprocals of the dodecagonal numbers is given by
∑ n = 1 ∞ 1 5 n 2 − 4 n = 5 16 ln ( 5 ) + 5 8 ln ( 1 + 5 2 ) + π 8 1 + 2 5 . {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{5n^{2}-4n}}={\frac {5}{16}}\ln \left(5\right)+{\frac {\sqrt {5}}{8}}\ln \left({\frac {1+{\sqrt {5}}}{2}}\right)+{\frac {\pi }{8}}{\sqrt {1+{\frac {2}{\sqrt {5}}}}}.}
See also Polygonal number Figurate number Dodecagon