A nonagonal number (or an enneagonal number) is a
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 term can mean
* polyg ...
that extends the concept of
triangular
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- collinea ...
and
square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The u ...
s to the
nonagon
In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon.
The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...
(a nine-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of nonagonal numbers are not rotationally symmetrical. Specifically, the ''n''th nonagonal number counts the number of dots in a pattern of ''n'' nested nonagons, all sharing a common corner, where the ''i''th nonagon in the pattern has sides made of ''i'' dots spaced one unit apart from each other. The nonagonal number for ''n'' is given by the formula:
:
Nonagonal numbers
The first few nonagonal numbers are:
:
0,
1,
9,
24,
46,
75,
111,
154
Year 154 ( CLIV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Lateranus (or, less frequently, year 907 ''Ab urbe cond ...
,
204
__NOTOC__
Year 204 ( CCIV) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Cilo and Flavius (or, less frequently, year 957 ''Ab urbe c ...
, 261, 325, 396, 474, 559, 651, 750, 856, 969,
1089
Year 1089 (Roman numerals, MLXXXIX) was a common year starting on Monday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* King Demetrius Zvonimir of Croatia dies after a 12-year reign, and is suc ...
, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364, 6666, 6975, 7291, 7614, 7944, 8281, 8625, 8976, 9334, 9699.
The
parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
of nonagonal numbers follows the pattern odd-odd-even-even.
Relationship between nonagonal and triangular numbers
Letting
denote the ''n''
th nonagonal number, and using the formula
for the ''n''
th triangular number
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 i ...
,
:
Test for nonagonal numbers
:
If is an integer, then is the -th nonagonal number. If is not an integer, then is not nonagonal.
See also
*
Centered nonagonal number
A centered nonagonal number (or centered enneagonal number) is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal ...
References
Figurate numbers
{{num-stub