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."

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).

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.

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

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

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.

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.

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.²¹

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

A Catalan factoriangular number is a number that is both a Catalan number and a factoriangular number. The only such numbers are 1, 2 and 5.⁴ They also proved in the same paper that the only central binomial coefficients which are also factoriangular numbers are 1 and 2.

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.