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Centered figurate number that represents a pentagon with a dot in the center
A centered pentagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers. The centered pentagonal number for n is given by the formula
P
n
=
5
n
2
−
5
n
+
2
2
,
n
≥
1
{\displaystyle P_{n}={{5n^{2}-5n+2} \over 2},n\geq 1}
The first few centered pentagonal numbers are
1 , 6 , 16 , 31 , 51 , 76 ,
106 , 141 , 181 , 226 , 276 ,
331 , 391 , 456 , 526 , 601 ,
681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976 (sequence A005891 OEIS ).
The parity of centered pentagonal numbers follows the pattern odd-even-even-odd, and in base 10 the units follow the pattern 1-6-6-1.
Centered pentagonal numbers follow the following recurrence relations :
P
n
=
P
n
−
1
+
5
n
,
P
0
=
1
{\displaystyle P_{n}=P_{n-1}+5n,P_{0}=1}
P
n
=
3
(
P
n
−
1
−
P
n
−
2
)
+
P
n
−
3
,
P
0
=
1
,
P
1
=
6
,
P
2
=
16
{\displaystyle P_{n}=3(P_{n-1}-P_{n-2})+P_{n-3},P_{0}=1,P_{1}=6,P_{2}=16}
P
n
=
5
T
n
−
1
+
1
{\displaystyle P_{n}=5T_{n-1}+1}
Possessing a specific set of other numbers
Expressible via specific sums
