Main Index Number Theory Sequences Recurrent sequences Linear recurrent sequences Figural numbers
  Subject Index
comment on the page

Triangular numbers

The typeset structureth triangular number typeset structure is defined as

T _ n = 1 + 2 + 3 + ... + (n - 1) + n .

We have

T _ n = n(n + 1)/2 = (n^2 + n)/2 = ((n + 1)/2)(1)

or the non-autonomous inhomogeneous recurrence

T _ n = T _ (n_ 1) + n .

The initial segment of triangular numbers starts with

The concept of the triangular number can be considered as an additive analog of the factorial.

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, ...

As a special case of figural numbers they are connected with a variety of other figural numbers. For instance, the sum of two consecutive triangular numbers is a square number

T _ n + T _ (n - 1) = (n^2/2 + n/2) + (n^2/2 - n/2) = n^2 .(2)

There is an alternative proof without words of this result:

[Graphics:HTMLFiles/TriangularNumbers_9.gif]

We also have

(T _ n - T _ (n - 1))^2 = n^2 .(3)

Every even perfect number is triangular. To see this write

2^(p - 1) M _ p = 2^(p - 1) (2^p - 1) = ((2^p)^2 - 2^p)/2 = (2^p (2^p - 1))/2 = ((M _ p + 1) M _ p)/2 = T _ M _ p ,

where typeset structure denotes a Mesrenne prime.

The sum of the reciprocals of all the triangular numbers is:

Underoverscript[∑, n = 1, arg3] 2/n(n + 1) = 2 Underoverscript[∑, n = 1, arg3] (1/n - 1/(n + 1)) = 2 .

On July 10, 1796  Carl Friedrich Gauß proved that every positive integer is representable as a sum of at most three triangular numbers. He recorded this discovery as  the 18th item in his famous diary in the form: "Heureka! num[erus]= Δ + Δ + Δ."  This result is actually a special case of Cauchy’s Polygonal Number Theorem .

Cite this web-page as:

Štefan Porubský: Triangular numbers.


Page created   .