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 (sequence A000567 in the OEIS)
Jun 12, 2016 The Sanwa octagonal gate is a must-have part in fighting games. While the 4-way gate that comes with standard issue joysticks serves it's purpose, the ability to lock-in moves with this 8-way gate makes it an essential purchase. Define octagonal. Octagonal synonyms, octagonal pronunciation, octagonal translation, English dictionary definition of octagonal. Having eight sides and eight angles. Octag′onally adv. Having eight sides and eight angles 2. Of or relating to an octagon ocˈtagonally adv.
Octagonal numbers can be formed by placing triangular numbers on the four sides of a square. To put it algebraically, the n-th octagonal number is
The octagonal number for n can also be calculated by adding the square of n to twice the (n - 1)th pronic number.
Octagonal numbers consistently alternate parity.
Octagonal numbers are occasionally referred to as 'star numbers,' though that term is more commonly used to refer to centered dodecagonal numbers.[1]
Sum of reciprocals[edit]
A formula for the sum of the reciprocals of the octagonal numbers is given by:[2]
Test for octagonal numbers[edit]
Solving the formula for the n-th octagonal number, for n gives
An arbitrary number x can be checked for octagonality by putting it in this equation. If n is an integer, then x is the n-th octagonal number. If n is not an integer, then x is not octagonal.
See also[edit]
References[edit]
- ^Deza, Elena; Deza, Michel (2012), Figurate Numbers, World Scientific, p. 57, ISBN9789814355483.
- ^Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers
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Over the octagonal window, too, such draperies fell in stately lines.
He sprang to the octagonal window, even as Dick took possession of the pistol.
The drum dome is octagonal, with eight ribs and as many windows.
On the other sides of the octagonal pavilion the other assistants are busy 'closing up.'
They are all round, though one belongs to an octagonal shaft.
It was octagonal in form, and was bordered all around with stone.
The octagonal story does not harmonize with the lower portion.
It was erected in the years 1432 and 1433, and was of octagonal shape.
It 15 is octagonal in plan, and possesses a fine groined roof.
The Font is often, following ancient custom, octagonal in form.