Centered triangle number

19 spheres in the form of nested triangles

A centered triangle number is a number that is based on the formula

${\displaystyle {\frac {3n^{2}-3n+2}{2}}}$

can be calculated from a natural number . The first triangular numbers centered are ${\displaystyle n}$

1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, ... (sequence A005448 in OEIS )

The centered triangular numbers , like the centered square numbers and the centered pentagonal and hexagonal numbers, belong to the centered polygonal numbers , i.e. to the flat figured numbers .

The centered triangular numbers indicate the number of pebbles to create a triangle according to the following rule: There is a pebble in the center and further pebbles are arranged around this in triangular layers with increasing sides. The number of stones in such an arrangement with layers is called the -th centered triangular number . ${\displaystyle n}$ ${\displaystyle (n+1)}$

For each centered triangular number can be represented as the sum of three consecutive normal triangular numbers. Furthermore, an integer division of any triangular number by 3 always results in the remainder 1 and the preceding triangular number as the quotient . ${\displaystyle n\geq 3}$ ${\displaystyle \Delta _{n-2}+\Delta _{n-1}+\Delta _{n}}$ ${\displaystyle ZD_{n}}$ ${\displaystyle ZD_{n-1}}$

The sum of the first triangular numbers ( ) gives the magic constant (line sum) of a magic square of the numbers 1 to . ${\displaystyle n}$ ${\displaystyle n\geq 3}$ ${\displaystyle n^{2}}$

Centered triangular primes

A centered triangular number that is prime is called a centered triangular prime number . The first centered triangular primes are:

19, 31, 109, 199, 409,… (Follow A125602 in OEIS )

See also

• Polygonal number

literature

• Lancelot Hogben : Mathematics for Everyone. An introduction to the science of numbers and figures. Newly revised edition. Pawlak, Herrsching 1985, ISBN 3-88199-208-1 , p. 151 ff.

Web links

• Eric W. Weisstein : centered triangular number . In: MathWorld (English).