**Centered Triangular Number**

A **centered** (or **centred**) **triangular number** is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. The centered triangular number for *n* is given by the formula

The following image shows the building of the centered triangular numbers using the associated figures: at each step the previous figure, shown in red, is surrounded by a triangle of new points, in blue.

The first few centered triangular numbers 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 OEIS).

Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. Also each centered triangular number has a remainder of 1 when divided by three and the quotient (if positive) is the previous regular triangular number.

The sum of the first *n* centered triangular numbers is the magic constant for an *n* by *n* normal magic square for *n* > 2.

Read more about Centered Triangular Number: Centered Triangular Prime

### Other articles related to "centered triangular number, centered triangular":

**Centered Triangular Number**- Centered Triangular Prime

... A

**centered triangular**prime is a

**centered triangular number**that is prime ... The first few

**centered triangular**primes are 19, 31, 109, 199, 409, … (sequence A125602 in OEIS) ...

### Famous quotes containing the words number and/or centered:

“No Government can be long secure without a formidable Opposition. It reduces their supporters to that tractable *number* which can be managed by the joint influences of fruition and hope. It offers vengeance to the discontented, and distinction to the ambitious; and employs the energies of aspiring spirits, who otherwise may prove traitors in a division or assassins in a debate.”

—Benjamin Disraeli (1804–1881)

“The difference between style and taste is never easy to define, but style tends to be *centered* on the social, and taste upon the individual. Style then works along axes of similarity to identify group membership, to relate to the social order; taste works within style to differentiate and construct the individual. Style speaks about social factors such as class, age, and other more flexible, less definable social formations; taste talks of the individual inflection of the social.”

—John Fiske (b. 1939)