A dodecagonal number is a figurate number

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 and different dimensions...

 that represents a dodecagon

Dodecagon

In geometry, a dodecagon is any polygon with twelve sides and twelve angles.- Regular dodecagon :It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°...

. The dodecagonal number for n is given by the formula 5n² - 4n, with n > 0. The first few dodecagonal numbers are:

1, 12

12 (number)

12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...

, 33

33 (number)

33 is the natural number following 32 and preceding 34.-In mathematics:33 is the largest positive integer that cannot be expressed as a sum of different triangular numbers. It is also the smallest odd repdigit that is not a prime number.33 is the eighth distinct semiprime comprising the prime...

, 64

64 (number)

64 is the natural number following 63 and preceding 65.-In mathematics:Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat...

, 105

105 (number)

105 is the natural number following 104 and preceding 106.-In mathematics:105 is a triangular number, a 12-gonal number and a Zeisel number. It is a sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7...

, 156

156 (number)

156 is the number equal to 100 + 50 + 6, following 155 and followed by 157.-In mathematics:It is a pronic number, a dodecagonal number, a refactorable number and a Harshad number.156 is a repdigit in base 5 .-In the military:...

, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729

1729 (number)

1729 is the natural number following 1728 and preceding 1730.1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan...

, 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 ...

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

Pronic number

A pronic number, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive integers, that is, n . The n-th pronic number is twice the n-th triangular number and n more than the n-th square number...

, or to put it algebraically, .

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

Fermat polygonal number theorem

In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive number can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of...

, every number is the sum of at most 12 dodecagonal numbers.