Triangular number

A triangular number is the sum of the n natural numbers from 1 to n. Triangular numbers are so called because they describe numbers of objects that can be arranged in a triangle. The nth triangular number is given by

T=1+2+3+\dotsb +(n-1)+n={\frac {n(n+1)}{2}}={\frac {n^{2}+n}{2}}={n+1 \choose 2}.

As shown in the rightmost term of this formula, every triangular number is a binomial coefficient: the nth triangular is the number of distinct pairs to be selected from n + 1 objects. In this form it solves the 'handshake problem' of counting the number of handshakes if each person in a room shakes hands once with each other person.

The sequence of triangular numbers (sequence A000217 in the OEIS) for n = 1, 2, 3... is:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

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Other properties

Every even perfect number is triangular, and no odd perfect numbers are known, hence all known perfect numbers are triangular.

In base 10, the digital root of a triangular number is always 1, 3, 6 or 9. Hence every triangular number is either divisible by three or has a remainder of 1 when divided by nine:

6 = 3×2,

10 = 9×1+1,

15 = 3×5,

21 = 3×7,

28 = 9×3+1,

...

The inverse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.

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

\!\ \sum _{n=1}^{\infty }{1 \over {{n^{2}+n} \over 2}}=2\sum _{n=1}^{\infty }{1 \over {n^{2}+n}}=2

This can be shown by using the basic sum of a telescoping series:

\!\ \sum _{n=1}^{\infty }{1 \over {n(n+1)}}=1

Two other interesting formulas regarding triangular numbers are:

T_{a+b}=T_{a}+T_{b}+ab

and

T_{ab}=T_{a}T_{b}+T_{a-1}T_{b-1},

both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.

In 1796, German mathematician and scientist Carl Friedrich Gauss discovered that every positive integer is representable as a sum of at most three triangular numbers, writing in his diary his famous words, "Heureka! num= Δ + Δ + Δ." Note that this theorem does not imply that the triangular numbers are different (as in the case of 20=10+10), nor that a solution with three nonzero triangular numbers must exist. This is a special case of Fermat's Polygonal Number Theorem.

Tests for triangular numbers

One can efficiently test whether a positive integer x is a triangular number by computing

n={\frac {{\sqrt {8x+1}}-1}{2}}.

If n is an integer, then x is the nth triangular number. If n is not an integer, then x is not triangular.

Alternate Way to Find the Sum of a Number Triangle

n={\frac {x+x+1}{2}}.

N is the sum of the current row, represented by x.

In non-mathematical format, multiply the current row by the following row and divide by 2. (i.e. row 6 is the current row, row 7 is the following row; 6 x 7 = 42, divided by 2 = 21, therefore the sum of the number of units in the triangle is 21)

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