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

pentagon In geometry, a pentagon (from the Greek language, 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 polygon, simple pentagon is ...

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,

106 106 may refer to: *106 (number), the number *AD 106, a year in the 2nd century AD *106 BC, a year in the 2nd century BC *106 (emergency telephone number), an Australian emergency number *106 (MBTA bus), a route of the Massachusetts Bay Transportatio ...

141 141 may refer to: * 141 (number), an integer * AD 141, a year of the Julian calendar * 141 BC __NOTOC__ Year 141 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Caepio and Pompeius (or, ...

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,

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 __NOTOC__ Year 391 ( CCCXCI) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Tatianus and Symmachus (or, less frequently, year 114 ...

456 __NOTOC__ Year 456 ( CDLVI) was a leap 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 Avitus without colleague (or, less frequently, year 1209 '' ...

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 ...

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