Centered octagonal number
Encyclopedia
A centered octagonal number is a centered
figurate number
that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. The centered octagonal number for n is given by the formula
where T is a regular triangular number
, or much more simply, by squaring the odd numbers:
The first few centered octagonal numbers are
1, 9, 25
, 49
, 81
, 121
, 169
, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089
All centered octagonal numbers are odd, and in base 10 one can notice the one's digits follow the pattern 1-9-5-9-1. An odd number is a centered octagonal number if and only if it is a perfect square
.
Calculating Ramanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number.
See also regular octagonal number.
Centered number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides...
figurate number
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 with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. The centered octagonal number for n is given by the formula
where T is a regular triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
, or much more simply, by squaring the odd numbers:
The first few centered octagonal numbers are
1, 9, 25
25 (number)
25 is the natural number following 24 and preceding 26.-In mathematics:It is a square number, being 5² = 5 × 5. It is the smallest square that is also a sum of two squares: 25 = 3² + 4²...
, 49
49 (number)
49 is the natural number following 48 and preceding 50.-In mathematics:* Forty-nine is the square of seven and is therefore the fourth squared prime number.The aliquot sum of forty-nine is 8...
, 81
81 (number)
81 is the natural number following 80 and preceding 82.-In mathematics:Eighty-one is the square of 9 and the fourth power of 3. Like all powers of three, 81 is a perfect totient number. It is a heptagonal number and a centered octagonal number. It is also a tribonacci number, and an open meandric...
, 121
121 (number)
121 is the natural number following 120 and preceding 122.-In mathematics:One hundred [and] twenty-one is a square and is the sum of three consecutive primes . There are no squares besides 121 known to be of the form 1 + p + p^2 + p^3 + p^4, where p is prime...
, 169
169 (number)
169 is the natural number following 168 and preceding 170.-In mathematics:* 169 is an odd number* 169 is a square number. 13 x 13 = 169, and the reverse 31 x 31 = 961...
, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089
1089 (number)
1089 is the integer after 1088 and before 1090.It is a square number , a nonagonal number, a 32-gonal number, a 364-gonal number, and a centered octagonal number.In base 10, the following steps always yield 1089:...
All centered octagonal numbers are odd, and in base 10 one can notice the one's digits follow the pattern 1-9-5-9-1. An odd number is a centered octagonal number if and only if it is a perfect square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
.
Calculating Ramanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number.
See also regular octagonal number.