Factoriangular number

In number theory, a factoriangular number is an integer formed by adding a factorial and a triangular number with the same index. The name is a portmanteau of "factorial" and "triangular."

Definition

For ${\displaystyle n\geq 1}$, the ${\displaystyle n}$th factoriangular number, denoted ${\displaystyle \operatorname {Ft} _{n}}$, is defined as the sum of the ${\displaystyle n}$th factorial and the ${\displaystyle n}$th triangular number:^[1]

${\displaystyle \operatorname {Ft} _{n}=n!+T_{n}=n!+{\frac {n(n+1)}{2}}}$.

The first few factoriangular numbers are:

${\displaystyle n}$	${\displaystyle n!}$	${\displaystyle T_{n}}$	${\displaystyle \operatorname {Ft} _{n}=n!+T_{n}}$
1	1	1	2
2	2	3	5
3	6	6	12
4	24	10	34
5	120	15	135
6	720	21	741
7	5,040	28	5,068
8	40,320	36	40,356
9	362,880	45	362,925
10	3,628,800	55	3,628,855

These numbers form the integer sequence A101292 in the Online Encyclopedia of Integer Sequences (OEIS).

Properties

Recurrence relations

Factoriangular numbers satisfy several recurrence relations. For ${\displaystyle n\geq 1}$,

${\displaystyle \operatorname {Ft} _{n+1}=(n+1)\left(\operatorname {Ft} _{n}-{\frac {n^{2}-2}{2}}\right)}$

And for ${\displaystyle n\geq 2}$,

${\displaystyle \operatorname {Ft} _{n}=n\left(\operatorname {Ft} _{n-1}-{\frac {n^{2}-2n-1}{2}}\right)}$

These are linear non-homogeneous recurrence relations with variable coefficients of order 1.

Generating functions

The exponential generating function ${\displaystyle E(x)=\sum _{n=0}^{\infty }\operatorname {Ft} _{n}{\tfrac {x^{n}}{n!}}}$ for factoriangular numbers is (for ${\displaystyle -1<x<1}$)

${\displaystyle E(x)={\frac {2+(2-5x^{2}+2x^{3}+x^{4})e^{x}}{2(1-x)^{2}}}}$

If the sequence is extended to include ${\displaystyle \operatorname {Ft} _{0}=1}$, then the exponential generating function becomes

${\displaystyle E(x)={\frac {2+(2x-x^{2}-x^{3})e^{x}}{2(1-x)}}}$.

Representations as sums of triangular numbers

Factoriangular numbers can sometimes be expressed as sums of two triangular numbers:

• ${\displaystyle \operatorname {Ft} _{n}=2T_{n}}$ if and only if ${\displaystyle n=1}$ or ${\displaystyle n=3}$.
• ${\displaystyle \operatorname {Ft} _{n}=T_{x}+T_{n}}$ if and only if ${\displaystyle 8n!+1}$ is a perfect square. For ${\displaystyle n\neq x}$, the only known solution is ${\displaystyle (\operatorname {Ft} _{5},T_{15})=(135,120)}$, giving ${\displaystyle \operatorname {Ft} _{5}=T_{5}+T_{15}}$.
• ${\displaystyle \operatorname {Ft} _{n}=T_{x}+T_{y}}$ if and only if ${\displaystyle 8\operatorname {Ft} _{n}+2}$ is a sum of two squares.

Representations as sums of squares

Some factoriangular numbers can be expressed as the sum of two squares. For ${\displaystyle n\leq 20}$, the factoriangular numbers that can be written as ${\displaystyle a^{2}+b^{2}}$ for some integers ${\displaystyle a}$ and ${\displaystyle b}$ include:

• ${\displaystyle \operatorname {Ft} _{1}=2=1^{2}+1^{2}}$
• ${\displaystyle \operatorname {Ft} _{2}=5=1^{2}+2^{2}}$
• ${\displaystyle \operatorname {Ft} _{4}=34=3^{2}+5^{2}}$
• ${\displaystyle \operatorname {Ft} _{9}=362,925=195^{2}+570^{2}}$

This result is related to the sum of two squares theorem, which states that a positive integer can be expressed as a sum of two squares if and only if its prime factorization contains no prime factor of the form ${\displaystyle 4k+3}$ raised to an odd power.

Fibonacci factoriangular numbers

A Fibonacci factoriangular number is a number that is both a Fibonacci number and a factoriangular number. There are exactly three such numbers:

• ${\displaystyle \operatorname {Ft} _{1}=2=F_{3}}$
• ${\displaystyle \operatorname {Ft} _{2}=5=F_{5}}$
• ${\displaystyle \operatorname {Ft} _{4}=34=F_{9}}$

This result was conjectured by Romer Castillo and later proved by Ruiz and Luca.^[2][1]

Pell factoriangular numbers

A Pell factoriangular number is a number that is both a Pell number and a factoriangular number.^[3] Luca and Gómez-Ruiz proved that there are exactly three such numbers: ${\displaystyle \operatorname {Ft} _{1}=2}$, ${\displaystyle \operatorname {Ft} _{2}=5}$, and ${\displaystyle \operatorname {Ft} _{3}=12}$.^[3]

Catalan factoriangular numbers

A Catalan factoriangular number is a number that is both a Catalan number and a factoriangular number.

Generalizations

The concept of factoriangular numbers can be generalized to ${\displaystyle (n,k)}$-factoriangular numbers, defined as ${\displaystyle \operatorname {Ft} _{n,k}=n!+T_{k}}$ where ${\displaystyle n}$ and ${\displaystyle k}$ are positive integers. The original factoriangular numbers correspond to the case where ${\displaystyle n=k}$. This generalization gives rise to factoriangular triangles, which are Pascal-like triangular arrays of numbers. Two such triangles can be formed:

• A triangle with entries ${\displaystyle \operatorname {Ft} _{n,k}}$ where ${\displaystyle k\leq n}$, yielding the sequence: 2, 3, 5, 7, 9, 12, 25, 27, 30, 34, ...
• A triangle with entries ${\displaystyle \operatorname {Ft} _{n,k}}$ where ${\displaystyle k\geq n}$, yielding the sequence: 2, 4, 5, 7, 8, 12, 11, 12, 16, 34, ...

In both cases, the diagonal entries (where ${\displaystyle n=k}$) correspond to the original factoriangular numbers.

See also

• Doubly triangular number
• Factorial prime
• Fibonacci number
• Lazy caterer's sequence
• Square triangular number