Centered triangular number

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title: "Centered triangular number" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["figurate-numbers"] description: "Centered figurate number that represents a triangle with a dot in the center" topic_path: "general/figurate-numbers" source: "https://en.wikipedia.org/wiki/Centered_triangular_number" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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).

::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/88/Centered_triangular_numbers.svg" caption="construction"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/3/36/centered_triangular_numbers_hex_grid.svg" caption="The first eight centered triangular numbers on a [[hex grid]]"] ::

Properties

• The gnomon of the n-th centered triangular number, corresponding to the (n + 1)-th triangular layer, is:

::C_{3,n+1} - C_{3,n} = 3(n+1).

• 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 (C3,n

: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, … .

The first simultaneously triangular and centered triangular numbers (C3,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),

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::