Is there a mathematical formula to determine how much XP is needed per level? In other words Is there a consistent formula for determining how much XP is needed to go up to any given level?
I originally thought it was the previous amount times 2, but that doesn't work for all levels.
I understand that there is a character advancement table, but I'm curious as to how they got those numbers and whether there is some mathematical formula to work it out.
WOTC has stated that the advancement rate is not constant across all levels. Rather, according to Unearthed Arcana: Three-Pillar Experience (August 2017), the standard XP and leveling system presented in the PHB
has faster advancement in a few spots [than the variant rules proposed in the UA.]
Note that although the quote comes from a UA playtest article, the quote itself is not playtest material -- it's the designers' commentary comparing the playtest material to the standard PHB rules.
And indeed if you look at the XP values required to advance from level to level, you won't be able to work out a straightforward mathematical formula for them, such as simple linearity or even exponentiality.
And the reasoning behind that less-than-straightforward system is clear: the designers wanted to make it faster to reach certain levels. For example, per Mike Mearls, the designers found through research that campaigns tend to stall at level 10, so they shortened the advancement track from 10 to 11.
After doing some interpolation I found that this equation was the best-fit (note \$x\$=level):
$$ \text{XP}=3.7825 x^4-134.59 x^3+2572.6 x^2-10699 x+10703 $$
However, it should be noted that this polynomial is unreliable until 5th level.
Due to the non-uniform level pacing detailed here and in screamline's answer, it's more productive to fit a curve to XP rewards by CR (since this doesn't depend on the current level of the party and is therefore invariant to level pacing) rather than than XP needed per level. It turns out that a good fit using round numbers is
$$\text{XP} \approx 50 \left(\text{CR} + 1\right)^2$$
Here's a semilog plot:
This is then modulated by the desired level pacing to get the XP requirements per level.
(However, it seems not as simple as the actual values being rounded from this equation. For example, it would be strange to round from 3200 to 2900 for CR 7. Also, the actual XP curve suddenly starts increasing rapidly above CR 20; it's best to consider that a separate regime entirely.)
I do not know of any direct designer statements that state that they intentionally chose a quadratic, let alone why they did so if they did. However, here's a possible explanation:
5e adopted the doctrine of bounded accuracy, where, according to designer Rodney Thompson:
The basic premise behind the bounded accuracy system is simple: we make no assumptions on the DM's side of the game that the player's attack and spell accuracy, or their defenses, increase as a result of gaining levels. Instead, we represent the difference in characters of various levels primarily through their hit points, the amount of damage they deal, and the various new abilities they have gained. Characters can fight tougher monsters not because they can finally hit them, but because their damage is sufficient to take a significant chunk out of the monster's hit points; likewise, the character can now stand up to a few hits from that monster without being killed easily, thanks to the character's increased hit points.
If damage and hit points both increase linearly with CR, their product increases quadratically. Granted, this does not take into account increases in attack bonus or AC, but if they play a lesser role then a quadratic may be a good-enough approximation.
In fact, under a Lanchester model of combat with a few additional assumptions, the estimated party resources consumed by an encounter is proportional to this product. It's a natural choice to award XP proportionally.
If you are using an Excel spreadsheet, you could use:
=CEILING(3.7825*Level^4-134.59*Level^3+2572.6*Level^2-10699*Level+10703,10^(LEN(ROUND(3.7825*Level^4-134.59*Level^3+2572.6*Level^2-10699*Level+10703,0))-2))
Exchange "Level" for the cell containing the Level value, and you would get a nice rounded value. But as noted in Nick's answer, the output for levels 1-4 does not reliably represent the desired XP values, but does pretty well from 5th level onward.
\begin{align} y ={}&\quad\ 5819.55617814588 - 11856.710519967924x + 7153.1988321984509x^2 \\& - 398.55230673086362x^3 - 1105.5094497604643x^4 + 462.03733563692845x^5 \\& - 80.142724143754521x^6 + 6.2081701486351601x^7 - 0.069480770175085027x^8 \\& - 0.016266981168801757x^9 + 0.0016558417135261892x^{10} \\& + 0.000069296325226162267x^{11} - 0.0000027175019434544397x^{12} \\& + 0.000000070646166571073877x^{13} - 0.0000000095150259370745423x^{14} \\& + 0.00000000028604436906456435x^{15} + 0.000000000012670491319511516x^{16} \\& - 0.00000000000032956267667046822x^{17} - 0.000000000000020151890236566283x^{18} \\& + 0.00000000000000056464447771396720x^{19} \end{align}
... does pretty well where x is your level and y is the XP requirement.
I applaud the much better answers to this question which attempt to fit a simpler curve; but just wanted to point out that you could fit a polynomial to this equation to get a perfectly accurate mathematical formula, but that doesn't mean you should! Such a formula is clearly even less useful than just trying to remember the list. Even this deliberately ridiculous polynomial is only accurate to a few percent. You need a lot more precision (or some more complex notation) to give you exactly what you want - but technically the answer is:
Yes; a mathematical formula to determine how much XP is needed for each level does exist. In fact there are infinitely many - and one such formula is just a more precise version of the one I've given above.
