Found some interesting equations:
- e * n! - [e * n!] for all n>0 ; The fractional or decimal remainder of the product of e! and n!, beyond the nearest integer value.
- Recursive Sequence ( u_n ) defined by ( u_{n+1} = -1 + (n+1)u_n ), starting with ( u_0 = e - 1 \
- Integral ( \int_{0}^{1} e^t (1-t)^n , dt \
- e (Γ(1 + n) - Γ(1 + n, 1)) (Another way of writing the previous)
| ( n ) | Recursive Sequence ( u_n ) defined by ( u_{n+1} = -1 + (n+1)u_n ), starting with ( u_0 = e - 1 ) | Integral ( \int_{0}^{1} e^t (1-t)^n , dt ) |
|---|---|---|
| 0 | ( e - 1 ) | ( e - 1 ) |
| 1 | ( e - 2 ) | ( e - 2 ) |
| 2 | ( 2e - 5 ) | ( 2e - 5 ) |
| 3 | ( 6e - 16 ) | ( 6e - 16 ) |
| 4 | ( 24e - 65 ) | ( 24e - 65 ) |
They output the factorial sequence (1, 1, 2, 6, 24 all multiplied by e) as well as the total number of permutations of all subsets of an n-set (A000522). A000522 is also related to the Number of operations of addition and multiplication needed to evaluate a determinant of order n by cofactor expansion (A000522-2)
The equation e (Γ(1 + n) - Γ(1 + n, 1)) reaches a limit at x=-1
In essence we can think of this, conceptually, as capturing the fractional growth beyond the integer milestones. In other words, it highlights the excess or remainder of growth that isn't encapsulated within the integer part
Starting Conditions: In other words, you can think of e-1 starting condition as the proxy for the full growth of something (any process that grows continuously at a constant rate.) for each time period, excluding the original "principal". In other words, what you get leftover when you perform a number of transformations.
Further, dividing this output by e yields an interesting concept.
∫₀¹ xⁿ e^{-x} dx
Take, for instance, n=3.
f(x) = x^3 e^{-x}
The derivative of the function ( f(x) = x^3 e^{-x} ) with respect to ( x ) is:
f'(x) = 3x^2 e^{-x} - x^3 e^{-x}
Conceptually, You're taking a quantity (in this case, ( x )), raising it to the third power to cube it, which significantly increases its value, especially for larger values of ( x ). Then you're applying an exponential decay to it with ( e^{-x} ), which shrinks it back down at a rate that increases as ( x ) gets larger. The integral measures how this process affects the area under the curve from the start of the interval at ( x = 0 ) to the end of the interval at ( x = 1 ). This gives you a single value that represents the combined effect of cubing and then exponentially decaying over the entire interval.