mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a centered pentagonal number is a centered

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathemat ...

that represents a

pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

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 Transpor ...

141 141 may refer to: * 141 (number), an integer * AD 141, a year of the Julian calendar * 141 BC, a year of the pre-Julian Roman calendar * 141 Lumen, a main-belt asteroid * Lockheed C-141 Starlifter The Lockheed C-141 Starlifter is a retired mil ...

181 Year 181 ( CLXXXI) was a common year starting on Sunday 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 condita''). The denomination 181 for this ye ...

226 Year 226 ( CCXXVI) was a common year starting on Sunday 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 condita''). The denomination 226 for this ...

, 276,

331 __NOTOC__ Year 331 ( CCCXXXI) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequently, year 1084 ''Ab urbe condita''). The denomination ...

, 391,

456 __NOTOC__ Year 456 ( CDLVI) was a leap year starting on Sunday 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 ''Ab urbe condita''). The denomination ...

526 __NOTOC__ Year 526 ( DXXVI) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Olybrius without colleague (or, less frequently, year 1279 ''Ab urbe condita''). The denomina ...

, 601, 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 In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

: :

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 A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

: :

P_=5T_+1

References

External links

* Figurate numbers {{num-stub

Properties

References

See also

External links