Taylor Series

My diurnal age today is 24491 and this number features in OEIS A077946:  

A077946

Expansion of $\dfrac{1}{1 - x - 2x^2 - 2x^3}$

The number arises as a coefficient of $x$ in the Taylor Series expansion of the function at $x=0$. The coefficients listed up to 24491 are:

1, 1, 3, 7, 15, 35, 79, 179, 407, 923, 2095, 4755, 10791, 24491

This means that the function can be expressed as:

$1 + x + 3x^2 + 7x^3 + 15x^4 + 35x^5 + \dots$

The Taylor Series for any continuous function $f(x)$at a point $x=a$ is given by the following expression: $$f(x)=f(a)+\frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{a!} (x-a)^2+ \frac{f'''(a)}{3!} (x-a)^3 + \dots$$ A Maclaurin Series is a Taylor Series where $a=0$.

Here is a permalink that will display these results using SageMathCell. I have a later post about Taylor Series that I created on April 27th 2018.

REFURBISHED on Saturday November 26th 2022