Octagonal number
Encyclopedia
An octagonal number is a figurate number
that represents an octagon. The octagonal number for n is given by the formula 3n2 - 2n, with n > 0. The first few octagonal numbers are:
1, 8, 21
, 40
, 65
, 96
, 133
, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936
Octagonal numbers can be formed by placing triangular numbers on the four sides of a square. To put it algebraically, the nth octagonal number is
which simplifies to the formulas given above.
The octagonal number for n can also be calculated by adding the square of n to twice the (n - 1)th pronic number
, or, to put it algebraically,
.
Octagonal numbers consistently alternate parity.
Octagonal numbers are occasionally referred to as "star number
s," though that term is more commonly used to refer to centered dodecagonal numbers.
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 an octagon. The octagonal number for n is given by the formula 3n2 - 2n, with n > 0. The first few octagonal numbers are:
1, 8, 21
21 (number)
21 is the natural number following 20 and preceding 22.-In mathematics:Twenty-one is the fifth discrete Semiprime and the second in the family. With 22 it forms the second discrete Semiprime pair...
, 40
40 (number)
40 is the natural number following 39 and preceding 41.Despite being related to the word "four" , 40 is spelled "forty", and not "fourty"...
, 65
65 (number)
65 is the natural number following 64 and preceding 66.-In mathematics:Sixty-five is the 23rd semiprime and the 3rd of the form it is an octagonal number. It is also a Cullen number...
, 96
96 (number)
96 is the natural number following 95 and preceding 97.-In mathematics:Ninety-six is an octagonal number, a refactorable number and an untouchable number. Since it is a multiple of 6, it is a semiperfect number...
, 133
133 (number)
133 is the natural number following 132 and preceding 134.-In mathematics:133 is an n whose divisors added up divide φ. It is an octagonal number and a Harshad number...
, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936
Octagonal numbers can be formed by placing triangular numbers on the four sides of a square. To put it algebraically, the nth octagonal number is
which simplifies to the formulas given above.
The octagonal number for n can also be calculated by adding the square of n to twice the (n - 1)th pronic number
Pronic number
A pronic number, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive integers, that is, n . The n-th pronic number is twice the n-th triangular number and n more than the n-th square number...
, or, to put it algebraically,
.Octagonal numbers consistently alternate parity.
Octagonal numbers are occasionally referred to as "star number
Star number
A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.The nth star number is given by the formula 6n + 1...
s," though that term is more commonly used to refer to centered dodecagonal numbers.