Termial

In mathematics, the termial of a positive integer n, denoted by n?, is the sum of all positive integers less than or equal to n. For example,

${\displaystyle 5?=5+4+3+2+1=15.}$

The value of 0? is 0, according to the convention for an empty sum.

The termial was coined by Donald E. Knuth in his The Art of Computer Programming. It is the additive analog of the factorial function, which is the product of integers from 1 to n. He used it to illustrate the extension of the domain from positive integers to the real numbers.^[1]

The termial of positive integers is also known as the triangular numbers.^[2] The first few (sequence A000217 in the OEIS) are

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...

Since the 18th century, Leonhard Euler and some other mathematicians had been trying to extend the domain of the factorial function to real numbers or even complex numbers, and eventually put forward the Gamma function.^[3] In 1997, Donald E. Knuth introduced the termial function n? in his The Art of Computer Programming, as an analog of factorial in addition, so as to illustrate the meaning of domain extension.^[1]

The termial function is defined by the sum

${\displaystyle n?=1+2+3+\cdots +(n-2)+(n-1)+n,}$

initially for integer n ≥ 1. This may be written in the Sigma sum notation as

${\displaystyle n?={\displaystyle \sum _{i=1}^n}i.}$

From these formulas, one may derive the recurrence relation

${\displaystyle n?=n+(n-1)?.}$

For example, one has

${\displaystyle \begin{array}{rl}5?& =5+4?\\ 6?& =6+5?\\ 50?& =50+49?\end{array}}$

and so on.

The termial function can be calculated using the summation formula for arithmetic sequence:

${\displaystyle n?=\frac{n(n+1)}{2}.}$

For example, ${\displaystyle 100?=\frac{100\times 101}{2}=5050}$.

In order for the recurrence relation to be extended to n = 0, it is necessary to define

${\displaystyle 0?=0}$

so that

${\displaystyle 1?=1+0?=1.}$

The termial function can also be defined for non-integer values using the formula ${\displaystyle n?=\frac{n(n+1)}{2}}$.

For example, ${\displaystyle (\frac{1}{2})?=\frac{3}{8}}$.

Termial is less frequently used in mathematics, but still it has some uses in fields such as combinatorics.

• For a set of n distinct elements, the number of 2-combination (i.e., number of ways to choose 2 of them) equals (n − 1)?. This is to say

${\displaystyle (\genfrac{}{}{0ex}{}{n}{2})=(n-1)?.}$

• In playing four fours, termial can be a useful tool to find the expression required, especially when the rules do not allow the use of decimal point and square root (which is because the numbers 0 and 2 are used invisibly). For example,

${\displaystyle 13=4?+4-4÷4}$

${\displaystyle 18=4!+4?-4\times 4}$

Similar to double factorial^[4], The sum of all the odd integers up to some odd positive integer n is called the double termial of n, and denoted by n??. That is,

${\displaystyle (2k-1)??={\displaystyle \sum _{i=1}^k}(2i-1)=1+3+5+\cdots +(2k-3)+(2k-1).}$

For example, ${\displaystyle 9??=1+3+5+7+9=25}$.

The sequence of double termial for n = 1, 3, 5, 7,... is the square number sequence.^[5] It starts as

1, 4, 9, 16, 25, 36, 49, ... (sequence A000290 in the OEIS)

Primial can be introduced as an analog of primorial, and denoted by n§. It is defined as the sum of prime numbers less than or equal to n^[6], i.e.

${\displaystyle n\mathrm{\S }=p_{\pi (n)}\mathrm{\S }={\displaystyle \sum _{i=1}^{\pi (n)}}{p}_i,}$

where ${\displaystyle \pi (n)}$ is the prime-counting function.

For example, ${\displaystyle 11\mathrm{\S }=p_5\mathrm{\S }=2+3+5+7+11=28}$.

The first few results are

0, 2, 5, 10, 17, 28, 41, ... (sequence A007504 in the OEIS)

Reciprocal termial is defined as the sum of reciprocal of first n positive integers. It is equal to the n-th harmonic number.^[7]

${\displaystyle {\displaystyle \sum _{i=1}^n}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}=H_n.}$

For example, ${\displaystyle {\displaystyle \sum _{i=1}^3}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}=\frac{11}{6}.}$

• Triangular number
• Summation
• Factorial
• Exponential factorial

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