Triangular number

A triangular number is one of the types of curly numbers , defined as the number of points that can be placed in the shape of a regular triangle (see figure). Obviously, from a purely arithmetic point of view, the nth triangular number is the sum of the n first natural numbers .

Sequence of triangular numbers $T_{n}$ ${\ displaystyle T_ {n}}$ $T_ {n}$ for $n=0,1,2,\ldots$ ${\ displaystyle n = 0,1,2, \ ldots}$ $n = 0,1,2, \ ldots$ starts like this:

0 , 1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , 45 , 55 , 66 , 78 , 91 , 105 , 120 ... (sequence A000217 in OEIS )

Content

Properties

Formulas for the n- th triangular number:
- $T_{n}={\frac {1}{2}}n(n+1)$ ${\ displaystyle T_ {n} = {\ frac {1} {2}} n (n + 1)}$ ;
- $T_{n}=1+2+3+\dots +(n-2)+(n-1)+n=\sum _{j=1}^{n}j$ ${\ displaystyle T_ {n} = 1 + 2 + 3 + \ dots + (n-2) + (n-1) + n = \ sum _ {j = 1} ^ {n} j}$ ;
- $T_{n}={n+1 \choose 2}$ ${\ displaystyle T_ {n} = {n + 1 \ choose 2}}$ - binomial coefficient .

For example, 1953 is the 62nd triangular number:

T_{62}={\frac {62\cdot 63}{2}}=1953.

{\ displaystyle T_ {62} = {\ frac {62 \ cdot 63} {2}} = 1953.}

The recurrence formula for the nth triangular number:
$T_{n}=T_{n-1}+n$ ${\ displaystyle T_ {n} = T_ {n-1} + n}$ .

Links between objects

If a $n$ ${\ displaystyle n}$ connected in pairs by segments, the number of segments will be expressed as a triangular number:

T_{n-1}={\frac {n(n-1)}{2}}

{\ displaystyle T_ {n-1} = {\ frac {n (n-1)} {2}}}

For example, if we have 4 objects, then we can only build

4\cdot 3/2=6

{\ displaystyle 4 \ cdot 3/2 = 6}

single connections between objects.

The sum of a finite series of triangular numbers is calculated by the formula:

S_{m-1}=1+3+6+\dots +{\frac {(m-1)m}{2}}={\frac {m^{3}-m}{6}}

{\ displaystyle S_ {m-1} = 1 + 3 + 6 + \ dots + {\ frac {(m-1) m} {2}} = {\ frac {m ^ {3} -m} {6} }}

.

A series of inverse triangular numbers converges:

1+{1 \over 3}+{1 \over 6}+{1 \over 10}+{1 \over 15}+\dots =2\sum _{n=1}^{\infty }\left({1 \over n}-{1 \over n+1}\right)=2

{\ displaystyle 1+ {1 \ over 3} + {1 \ over 6} + {1 \ over 10} + {1 \ over 15} + \ dots = 2 \ sum _ {n = 1} ^ {\ infty} \ left ({1 \ over n} - {1 \ over n + 1} \ right) = 2}

The mysterious “ number of the beast ” (666) is the 36th triangular. It is the smallest triangular number, which is representable as the sum of the squares of triangular numbers ^[1] : $666=15^{2}+21^{2}.$ ${\ displaystyle 666 = 15 ^ {2} + 21 ^ {2}.}$
The third line (diagonal) of the Pascal triangle consists of triangular numbers.

Relationship with other classes of numbers

The sum of two consecutive triangular numbers is a square number (full square), i.e.

T_{n-1}+T_{n}=n^{2}

{\ displaystyle T_ {n-1} + T_ {n} = n ^ {2}}

.

Examples:

6 + 10 = 16

10 + 15 = 25

Each even perfect number is triangular ^[2] .

Any natural number can be represented as the sum of no more than three triangular numbers. The statement was first formulated in 1638 by Pierre Fermat in a letter to Mersenne without proof, first proved in 1796 by Gauss ^[3] .

Natural number $m$ ${\ displaystyle m}$ is triangular if and only if the number $8m+1$ ${\ displaystyle 8m + 1}$ is a full square . In fact, if $m$ ${\ displaystyle m}$ triangular then $8m+1=8{\frac {n(n+1)}{2}}+1=4n^{2}+4n+1=(2n+1)^{2}.$ ${\ displaystyle 8m + 1 = 8 {\ frac {n (n + 1)} {2}} + 1 = 4n ^ {2} + 4n + 1 = (2n + 1) ^ {2}.}$ Conversely, the number $8m+1$ ${\ displaystyle 8m + 1}$ odd, and if it is equal to the square of some number $a,$ ${\ displaystyle a,}$ then $a$ ${\ displaystyle a}$ also odd: $a=2c+1,$ ${\ displaystyle a = 2c + 1,}$ and we get the equality: $8m+1=(2c+1)^{2}=4c^{2}+4c+1,$ ${\ displaystyle 8m + 1 = (2c + 1) ^ {2} = 4c ^ {2} + 4c + 1,}$ where from: $m={\frac {c(c+1)}{2}}$ ${\ displaystyle m = {\ frac {c (c + 1)} {2}}}$ Is a triangular number.

The square of the nth triangular number is the sum of the cubes of the n first natural numbers ^[4] .

There are infinitely many triangular numbers that are simultaneously square (“ square triangular numbers ”) ^[5] : $1,36,1225,41616,1413721\dots$ ${\ displaystyle 1.36.1225.41616.1413721 \ dots}$ (sequence A001110 in OEIS ).

Variations and generalizations

The concept of a flat triangular number can be generalized to three or more dimensions. Their spatial counterpart are tetrahedral numbers , and in an arbitrary $d$ ${\ displaystyle d}$ -dimensional space, one can define hypertetrahedral numbers :

T_{n}^{[d]}={\frac {(n-1+d)!}{(n-1)!\ d!}}

{\ displaystyle T_ {n} ^ {[d]} = {\ frac {(n-1 + d)!} {(n-1)! \ d!}}}

Their special case are:

$T_{n}^{[2]}$ ${\ displaystyle T_ {n} ^ {[2]}}$ Are triangular numbers.
$T_{n}^{[3]}$ ${\ displaystyle T_ {n} ^ {[3]}}$ - tetrahedral numbers.
$T_{n}^{[4]}$ ${\ displaystyle T_ {n} ^ {[4]}}$ - pentatope numbers .

Notes

↑ Desa E., Desa M., 2016 , p. 225.
↑ Voight, John. Perfect numbers: an elementary introduction // University of California, Berkley. - 1998 .-- S. 7 .
↑ Desa E., Desa M., 2016 , p. ten.
↑ Desa E., Desa M., 2016 , p. 79.
↑ Desa E., Desa M., 2016 , p. 25–33.

Literature

Vilenkin N. Ya., Shibasov L. P. Shibasova 3. F. Behind the pages of a mathematics textbook: Arithmetic. Algebra. Geometry. - M .: Education, 1996.- S. 30. - 320 p. - ISBN 5-09-006575-6 .
Gleizer G.I. History of mathematics at school . - M .: Enlightenment, 1964 .-- 376 p.
Desa E., Desa M. Curly numbers. - M .: ICMMO, 2016 .-- 349 p. - ISBN 978-5-4439-2400-7 .

Links

Curly numbers
Figurate Numbers on MathWorld

[_7673ebf0325ed2c0-1] Desa E., Desa M., 2016 , p. 225.

[2] Voight, John. Perfect numbers: an elementary introduction // University of California, Berkley. - 1998 .-- S. 7 .

[_cef8a15096c5ef5c-3] Desa E., Desa M., 2016 , p. ten.

[_cef89f5096c5ec3f-4] Desa E., Desa M., 2016 , p. 79.

[_72c0946749498450-5] Desa E., Desa M., 2016 , p. 25–33.