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Centered triangular number
Centered figurate number that represents a triangle with a dot in the center
Centered figurate number that represents a triangle with a dot in the center
A centered (or centred) triangular number is a 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 n.
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).
construction
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:
::C_{3,n} = 1 + 3 \frac{n(n+1)}{2} = \frac{3n^2 + 3n + 2}{2}.
- 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 numbers.
- 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:
:C_{3,n} = \frac{3C_{4,n} + 1}{4},
where
:C_{4,n} = n^{2} + (n+1)^{2}.
## Lists of centered triangular numbers
The first centered triangular numbers (*C*3,*n*
:[1](1-number), [4](4-number), [10](10-number), [19](19-number), [31](31-number), [46](46-number), [64](64-number), [85](85-number), [109](109-number), [136](136-number), [166](166-number), [199](199-number), [235](235-number), [274](274-number), 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* 9) are:
:1, 10, 136, 1 891, 26 335, 366 796, 5 108 806, 71 156 485, 991 081 981, … .
## 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 |x| , in which case it can be expressed as the meromorphic generating function
: 1 + 4x + 10x^2 + 19x^3 + 31x^4 +~... = \frac{1-x^3}{(1-x)^4} = \frac{x^2+x+1}{(1-x)^3} ~.
## References
- Lancelot Hogben: *Mathematics for the Million* (1936), republished by W. W. Norton & Company (September 1993),
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::callout[type=info title="Wikipedia Source"]
This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Centered_triangular_number) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Centered_triangular_number?action=history).
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figurate-numbers
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