View source for Centered triangular number

{{Short description|Centered figurate number that represents a triangle with a dot in the center}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
{{no footnotes|date=June 2014}}

A '''centered''' (or '''centred''') '''triangular number''' is a [[Centered number|centered]] [[figurate number]] that represents an [[equilateral triangle]] with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers.

This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point
is less than or equal to <math>n</math>.

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).

[[File:Centered triangular numbers.svg|946x946px|construction]]
[[File:centered_triangular_numbers_hex_grid.svg|thumb|The first eight centered triangular numbers on a [[hex grid]] ]]
==Properties==
*The [[Gnomon (figure)|gnomon]] of the ''n''-th centered triangular number, corresponding to the (''n'' + 1)-th triangular layer, is:

::<math>C_{3,n+1} - C_{3,n} = 3(n+1).</math>

*The ''n''-th centered triangular number, corresponding to ''n'' layers ''plus'' the center, is given by the formula:

::<math>C_{3,n} = 1 + 3 \frac{n(n+1)}{2} = \frac{3n^2 + 3n + 2}{2}.</math>

*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]]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:

:<math>C_{3,n} = \frac{3C_{4,n} + 1}{4},</math>

where

:<math>C_{4,n} = n^{2} + (n+1)^{2}.</math>

==Lists of centered triangular numbers==
The first centered triangular numbers (''C''<sub>3,''n''</sub> < 3000) are:

:[[1 (number)|1]], [[4 (number)|4]], [[10 (number)|10]], [[19 (number)|19]], [[31 (number)|31]], [[46 (number)|46]], [[64 (number)|64]], [[85 (number)|85]], [[109 (number)|109]], [[136 (number)|136]], [[166 (number)|166]], [[199 (number)|199]], [[235 (number)|235]], [[274 (number)|274]], 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, … {{OEIS|id=A005448}}.

The first simultaneously triangular and centered triangular numbers (''C''<sub>3,''n''</sub> = ''T''<sub>''N''</sub> < 10<sup>9</sup>) are:

:1, 10, 136, 1 891, 26 335, 366 796, 5 108 806, 71 156 485, 991 081 981, … {{OEIS|id=A128862}}.

==The generating function==
If the centered triangular numbers are treated as the coefficients of 
the McLaurin series of a function, that function converges for all <math> |x| < 1</math>, in which case it can be expressed as the meromorphic generating function
:<math> 1 + 4x + 10x^2 + 19x^3 + 31x^4 +~... = \frac{1-x^3}{(1-x)^4} = \frac{x^2+x+1}{(1-x)^3} ~.</math>

==References==
*[[Lancelot Hogben]]: ''Mathematics for the Million'' (1936), republished by W. W. Norton & Company (September 1993), {{ISBN|978-0-393-31071-9}}
*{{MathWorld|urlname=CenteredTriangularNumber|title=Centered Triangular Number}}

{{Figurate numbers}}

[[Category:Figurate numbers]]