A centered (or centred) triangular number is a
centered 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 represents an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers.
The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue).
Properties
*The
gnomon of the ''n''-th centered triangular number, corresponding to the (''n'' + 1)-th triangular layer, is:
::
*The ''n''-th centered triangular number, corresponding to ''n'' layers ''plus'' the center, is given by the formula:
::
*Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if positive) is the previous regular triangular number.
*Each centered triangular number from 10 onwards is the sum of three consecutive regular
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 ...
s.
*For ''n'' > 2, the sum of the first ''n'' centered triangular numbers is the
magic constant for an ''n'' by ''n'' normal
magic square.
Relationship with centered square numbers
The centered triangular numbers can be expressed in terms of the centered square numbers:
:
where
:
Lists of centered triangular numbers
The first centered triangular numbers (''C''
3,''n'' < 3000) are:
:
1,
4,
10,
19,
31,
46,
64,
85,
109,
136 136 may refer to:
*136 (number)
*AD 136
*136 BC 136 may refer to:
*136 (number)
*AD 136
Year 136 ( CXXXVI) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar, the 136th Year of the Common Era (C ...
,
166,
199
Year 199 ( CXCIX) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was sometimes known as year 952 '' Ab urbe condita''. The denomination 199 for this year has been used since the ...
,
235,
274
Year 274 ( CCLXXIV) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelianus and Capitolinus (or, less frequently, year 1027 '' ...
, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, … .
The first simultaneously triangular and centered triangular numbers (''C''
3,''n'' = ''T''
''N'' < 10
9) are:
:1, 10, 136, 1 891, 26 335, 366 796, 5 108 806, 71 156 485, 991 081 981, … .
The generating function
The generating function that gives the centered triangular numbers is:
:
References
*
Lancelot Hogben
Lancelot Thomas Hogben FRS FRSE (9 December 1895 – 22 August 1975) was a British experimental zoologist and medical statistician. He developed the African clawed frog ''(Xenopus laevis)'' as a model organism for biological research in his ear ...
: ''Mathematics for the Million'' (1936), republished by W. W. Norton & Company (September 1993),
*
{{Figurate numbers
Figurate numbers