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Inmathematics, adodecagonal number is afigurate number that represents adodecagon. The dodecagonal number forn is given by the formula

${\displaystyle D_n=5n^2-4n}$

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, ... (sequenceA051624 in theOEIS)

• The dodecagonal number forn can be calculated by adding the square ofn to four times the (n - 1)thpronic number, or to put it algebraically,${\displaystyle D_n=n^2+4(n^2-n)}$.

• Dodecagonal numbers consistently alternateparity, and in base 10, their units place digits follow the pattern 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

• By theFermat polygonal number theorem, every number is the sum of at most 12 dodecagonal numbers.

• ${\displaystyle D_n}$ is the sum of the first n natural numbers congruent to 1 mod 10.

• ${\displaystyle D_{n+1}}$ is the sum of all odd numbers from 4n+1 to 6n+1.

A formula for thesum of the reciprocals of the dodecagonal numbers is given by$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{5n^2-4n}=\frac{5}{16}\ln \left(5\right)+\frac{\sqrt{5}}{8}\ln \left(\frac{1+\sqrt{5}}{2}\right)+\frac{\pi }{8}\sqrt{1+\frac{2}{\sqrt{5}}}.}$$

• Polygonal number
• Figurate number
• Dodecagon

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