Centered decagonal number

A centered decagonal number is a centered figurate number that represents a decagon 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

$(5(n^2-n))+1$

Thus, the first few centered decagonal numbers are

1, 11, 31, 61, 101, 151, 211, 281, 361, 451,551, 661, 781, 911, 1051, ... OEIS|id=A062786

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

$CD\_n\; =\; CD\_\{n-1\}+10(n-1)$

where "CD"₁ is 1.

Centered decagonal prime

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