A centered pentagonal number is a centered figurate number that represents a

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...

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\geq1

The first few centered pentagonal numbers are 1, 6, 16, 31, 51, 76, 106, 141,

181 Year 181 ( CLXXXI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Burrus (or, less frequently, year 934 ''Ab urbe condit ...

226 Year 226 ( CCXXVI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Marcellus (or, less frequently, year 979 ''Ab urbe cond ...

276 __NOTOC__ Year 276 ( CCLXXVI) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Tacitus and Aemilianus (or, less frequently, year 1029 ...

, 331, 391, 456,

526 __NOTOC__ Year 526 ( DXXVI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Olybrius without colleague (or, less frequently, year 1 ...

601 __NOTOC__ Year 601 ( DCI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. The denomination 601 for this year has been used since the early medieval period, when the Anno Domini calendar era bec ...

, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976 .

Properties

*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_=P_+5n , P_0=1

P_=3(P_-P_)+P_ , P_0=1,P_1=6,P_2=16

*Centered pentagonal numbers can be expressed using Triangular Numbers: :

P_=5T_+1

External links

* Figurate numbers {{num-stub

Properties

See also

External links