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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 .
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).
![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Centered_triangular_numbers_hex_grid.svg/220px-Centered_triangular_numbers_hex_grid.svg.png)
Properties
edit- 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 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
editThe centered triangular numbers can be expressed in terms of the centered square numbers:
where
Lists of centered triangular numbers
editThe first centered triangular numbers (C3,n < 3000) are:
- 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 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, … (sequence A005448 in the OEIS).
The first simultaneously triangular and centered triangular numbers (C3,n = TN < 109) are:
The generating function
editIf the centered triangular numbers are treated as the coefficients of the McLaurin series of a function, that function converges for all , in which case it can be expressed as the meromorphic generating function
References
edit- Lancelot Hogben: Mathematics for the Million (1936), republished by W. W. Norton & Company (September 1993), ISBN 978-0-393-31071-9
- Weisstein, Eric W. "Centered Triangular Number". MathWorld.