In mathematics, a centered dodecahedral number is a centered figurate number that represents a dodecahedron. The centered dodecahedral number for a specific n is given by
( 2 n + 1 ) ( 5 n 2 + 5 n + 1 ) {\displaystyle (2n+1)\left(5n^{2}+5n+1\right)}
The first such numbers are: 1, 33, 155, 427, 909, 1661, 2743, 4215, 6137, 8569, … (sequence A005904 in the OEIS).
Congruence Relations
C D C ( n ) ≡ 1 ( mod 2 ) {\displaystyle CDC(n)\equiv 1{\pmod {2}}}
C D C ( n ) ≡ 1 − n ( mod 3 ) {\displaystyle CDC(n)\equiv 1-n{\pmod {3}}}
C D C ( n ) ≡ 2 n + 1 ( mod 3 , 5 , 6 , 10 ) {\displaystyle CDC(n)\equiv 2n+1{\pmod {3,5,6,10}}}