Dodecagonal number (original) (raw)
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Figurate number representing a dodecagon
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)
- 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.
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}}}}}.} 