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 ancient Greek mathemat ...

that extends the concept of

triangular A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...

and

square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...

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 word, hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in Fre ...

(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 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: :

\frac

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 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 condita''). The denomination 154 for this ...

, 204,

261 __NOTOC__ Year 261 (Roman numerals, CCLXI) was a common year starting on Tuesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Gallienus and Taurus (or, less frequently, year 1014 ''Ab urbe condita''). The ...

, 325, 396, 474, 559, 651,

750 __NOTOC__ Year 750 ( DCCL) was a common year starting on Thursday of the Julian calendar, the 750th year of the Common Era (CE) and Anno Domini (AD) designations, the 750th year of the 1st millennium, the 50th year of the 8th century, and the ...

856 __NOTOC__ Year 856 ( DCCCLVI) was a leap year starting on Wednesday of the Julian calendar. Events By place Byzantine Empire * March 15 – Emperor Michael III overthrows the regency of his mother Theodora. He appoints his uncle ...

969 Year 969 ( CMLXIX) was a common year starting on Friday of the Julian calendar, the 969th year of the Common Era (CE) and ''Anno Domini'' (AD) designations, the 969th year of the 1st millennium, the 69th year of the 10th century, and the 10th ...

, 1089, 1216,

1350 Year 1350 ( MCCCL) was a common year starting on Friday of the Julian calendar. Events January–December * January 9 – Giovanni II Valente becomes Doge of Genoa. * May 23 (possible date) – Hook and Cod wars in the Cou ...

, 1491,

1639 Events January–March * January 19 – Hämeenlinna () is granted privileges, after it separates from the Vanaja parish, as its own city in Tavastia. *c. January – The first printing press in British North America is ...

1794 Events January–March * January 1 – The Stibo Group is founded by Niels Lund as a printing company in Aarhus (Denmark). * January 13 – The U.S. Congress enacts a law providing for, effective May 1, 1795, a United St ...

1956 Events January * January 1 – The Anglo-Egyptian Sudan, Anglo-Egyptian Condominium ends in Sudan after 57 years. * January 8 – Operation Auca: Five U.S. evangelical Christian Missionary, missionaries, Nate Saint, Roger Youderian, E ...

, 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 of nonagonal numbers follows the pattern odd-odd-even-even.

Relationship between nonagonal and triangular numbers

Letting

N_n

denote the ''n''^th nonagonal number, and using the formula

T_n = \frac

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 in ...

, :

7N_n + 3 = T_

Test for nonagonal numbers

\mathsf~x = \frac

. If is an integer, then is the -th nonagonal number. If is not an integer, then is not nonagonal.

References

Figurate numbers {{num-stub

Nonagonal numbers

Relationship between nonagonal and triangular numbers

Test for nonagonal numbers

See also

References