Centered pentagonal number
Encyclopedia
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 and different dimensions...
that represents a pentagon
Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
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
The first few centered pentagonal numbers are
1, 6, 16
16 (number)
16 is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and ....
, 31
31 (number)
31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...
, 51, 76
76 (number)
76 is the natural number following 75 and preceding 77.-In mathematics:Seventy-six is a Lucas number, an automorphic number, a nontotient, a 14-gonal number, and a centered pentagonal number....
, 106
106 (number)
106 is the natural number following 105 and preceding 107.-In mathematics:106 is the thirty-first distinct biprime and the fifteenth of the form...
, 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 .
The parity of centered pentagonal numbers follows the pattern even-even-odd-odd, and in base 10 the units follow the pattern
6-6-1-1.