Centered figurate number that represents a pentagon with a dot in the center
Inmathematics , acentered pentagonal number is acentered figurate number that represents apentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers.[ 1] 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 (sequenceA005891 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 followingrecurrence 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
