A centered decagonal number is a centered

Centered number

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides...

 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 decagon

Decagon

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°...

 with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for n is given by the formula

Thus, the first few centered decagonal numbers are

1, 11

11 (number)

11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...

, 31

31 (number)

31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...

, 61

61 (number)

61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 151

151 (number)

151 is a natural number. It follows 150 and precedes 152.-In mathematics:151 is the 36th prime number, the previous is 149, with which it comprises a twin prime. 151 is also a palindromic prime. 151 is a centered decagonal number...

, 211, 281, 361, 451, 551, 661, 781, 911

911 (number)

911 is the number following 910 and preceding 912.It is a prime number, a Sophie Germain prime and the sum of three consecutive primes . It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. Since 913 is a semiprime, 911 is a Chen prime...

, 1051, ...

Like any other centered k-gonal number, the nth centered decagonal number can reckoned by multiplying the (n − 1)th triangular number

Triangular number

A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

 by k, 10 in this case, then adding 1. As a consequence of performing the calculation in base 10, the centered decagonal numbers can be obtained by simply adding a 1 to the right of each triangular number. Therefore, all centered decagonal numbers are odd and in base 10 always end in 1.

Another consequence of this relation to triangular numbers is the simple recurrence relation

Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....

 for centered decagonal numbers

where CD₁ is 1.

Centered decagonal prime

A centered decagonal prime is a centered decagonal number that is prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

. The first few centered decagonal primes are

11, 31, 61, 101, 151, 211, 281, 661, 911, 1051, 1201, 1361, 1531, 1901, 2311, 2531, 3001, 3251, 3511, 4651, 5281, ....

See also regular decagonal number

Decagonal number

A decagonal number is a figurate number that represents a decagon. The n-th decagonal number is given by the formulaThe first few decagonal numbers are:...

.