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Factoriangular Numbers, coined by Romer C. Castillo, are the sums of the factorial and triangular numbers for a given n. For example, for n=3 the factorial number is 3!=6 and the triangular number is 1+2+3=6. Thus, the factoriangular number is 6+6=12.

The sequence of factoriangular numbers (sequence A101292 in the OEIS) starting at the 0th factoriangular number is 1, 2, 5, 12, 34, 135, 741, 5068, 40356, 362925, 3628855, 39916866, 479001678, 6227020891, 87178291305, 1307674368120, 20922789888136, 355687428096153, 6402373705728171, 121645100408832190, 2432902008176640210, 51090942171709440231, 1124000727777607680253...

The formula for the positive nth factoriangular number (Ftn) is given by $$Ft_n=n!+\frac{n(n+1)}{2}$$ where $$n!$$ is the factorial function and $$\frac{n(n+1)}{2}$$ is the explicit formula for triangular numbers.

Properties
For n≥2, Ftn is even if n is a nonzero multiple of 4 or 3 greater than a non-zero multiple of 4. Ftn is odd if n is 1 greater than a non-zero multiple of 4 or if n is 2 greater than a multiple of 4.

For n≥3, Ftn is composite

An odd Ftn is divisible by n if n is odd and by n/2 if n is even

For even n ≥ 2, there is a positive integer k such that $$Ft_n=[\frac{Ft_n+1}{n+1}]+k$$ with $$k=\frac{n^2-2}{2}$$ and for odd n≥3 there is a positive integer k such that $$2Ft_n=[\frac{2Ft_n+1}{n+1}]+k$$ and $$k=n^2-2$$

Recurrence relations
Factoriangular numbers follow the below recurrence relations. For n≥1 $$Ft_{n+1}=(n+1)(Ft_n-\frac{n^2-2}{2})$$ For n≥2 $$Ft_n=n(Ft_{n-1}-\frac{n^2-2n-1}{2})$$

Moment generating functions
For n≥1 and -1=2 the sum of the first and last terms of this runsum is twice the nth factoriangular number divided by n.

The nth factoriangular number can also be written as $$Ft_n=T_{(n-1)!+n}-T_{(n-1)!}$$

Factorial and triangular numbers are related by $$n!=T_{(n-1)!+n}-T_{(n-1)!}-T_n$$

Except for Ft1=2 all factoriangular numbers are trapezoidal (the sum of at least two consecutive positive integers) and polite numbers (a positive integer that can be written as the sum of two or more consecutive positive integers. A polite number is considered trapezoidal if all numbers in the sum are greater than one)

No factoriangular number greater than 2 is a power of 2.

Other properties
For n≥1, only the pairs Ft1=2 and T1=1, and Ft3=12 and T3=6 satisfy the relation Ftn=2Tn

For n,x ≥1 $$Ft_n=2T_x$$ if and only if 4Ftn+1 is a square

For n,x ≥1 $$Ft_n=T_x+T_n$$ if and only if 8n!+1 is a square

For n,x,y ≥1 $$Ft_n=T_x+T_y$$ if and only if 8Ftn+2 is the sum of two squares

Conjectures
There is no factoriangular number that is a perfect number.

Except for 2, 5, and 12, there is no factoriangular number that is a divisor of another.

2, 5, and 34 are the only factoriangular numbers that are Fibonacci numbers.

There is no factoriangular number that has a Zeckendorf's decomposition of only 2 terms.

Only 12, 135, 741, and 5068 has a Zeckendorf's decomposition of only 3, 4, 5, 6 terms respectively.

For n,x ≥1 and n≠x there is no pair of factoriangular numbers and triangular numbers that satisfy the relation Ftn=2Tx.

Ft1=2 and Ft3=12 are the only factoriangular numbers that are twice a triangular number.

For n,x ≥1 and n≠x Ftn is the nth factoriangular number and Ti is the ith triangular number the only solution for $$Ft_n=T_x+T_n$$ is n=5 and n=15.

Open questions
Is there any factoriangular number that is also triangular other than Ft6=T38=741?

Other than Ft1=2 and Ft4=34 is there any other factoriangular number that can be expressed both as the sum of two triangular numbers and as the sum of two squares?