I track the behavior of random numbers and I have discovered that once a number appears, it tends to reappear again shortly thereafter. For example, I've been tracking the Red Powerball in the Powerball lottery, it's the single red ball from a pool of $1$ through $26$ numbers.
The plot below is a gap chart or plot ... a consolidation of the distance between when a number is drawn and the next time it is drawn again. For example, at position $1$ on the horizontal axis, indicates that $37$ times balls repeat on the next drawing. Positions $2$ and $3$ on the horizontal axis indicate that balls have appeared $40$ times once again $2$ and $3$ draws later respectively. As we move toward the right, we see the lengths of gaps between repeats increase gradually. At the very far right, there was a gap of a ball repeating after 198 draws, that's a drought.
A Suspect Chart is also included to indicate a gap chart that would indicate manufactured repeats at locations $44$ through $50$.
Is there a statistical explanation for this phenomena?
The number of games (aka draws) represented in the Powerball plot is $969$ and the number of draws in the Megemilions plot is $605$.
A chart of a Suspect Drawing. If one were to see this when analyzing gap analysis, then one could assert that the drawing was not random?
Also added for completeness, is the Megamilions Blue Ball MB, which has the range $1$ through $25$. It also indicates that when any given MB is drawn, it's likely to be drawn again within the next few draws.
It would be nice if I could attach text data to this question for reviewers to download.
Here is an example of $35$ random numbers (pool size $1$ through $10$) drawn in an Excel spreadsheet taken out to gap of $5$. Horizontal position $1$ in my plot is gap$0$, horizontal position $2$ in my plot is gap$1$, ... and so on.
There are two different kinds of "gap" that can be studied, which I'll call Type 1 and Type 2; e.g., if $\ 8XX...X8\ $ occurs in the sequence of draws:
Type 1: There's a gap between these two 8s no matter what occurs between them.
Type 2: There's a gap between these two 8s only if no 8 occurs between them.
Your original description and your added Excel worksheet -- as well as my original answer -- correspond to Type 1. (Unlike your worksheet, Type 2 would have at most one gap-entry for each draw, i.e. for each row.) On the other hand, all of your plots seem to show Type 2 analysis, which results in a Geometric distribution.
Also, note that the "size" of a gap can be taken either as the number of draws before a repeat occurs (as you've done), or it can be the number of draws including the repeat (as I have done).
"Type 1" gap analysis
According to that description (and your Excel worksheet), given a specific gap (say $\delta$), the bar at that position is showing the number of times $X_i=X_{i+\delta}$ occurs with $i\in\{1,...,n-\delta\}$, where $n$ is the total number of draws and $X_i$ is the $i$th draw. But in that case, none of your charts resemble what's expected if the draws are independent and uniformly distributed on $\{1,...,26\}$.
Let the sequence of Red Powerballs be $X_1,X_2,...X_n$, and let $g(\delta)$ be the number of times a gap of size $\delta$ occurs. Then using indicator functions (Iverson brackets $[...]$) we have:
$$\begin{align}g(\delta) &=\sum_{i=1}^{n-\delta}[X_i=X_{i+\delta}]\\[2ex] E(g(\delta))&=\sum_{i=1}^{n-\delta}P[X_i=X_{i+\delta}]\\[1ex] &={n-\delta\over 26}\\[2ex] V(g(\delta))&=E(g(\delta)^2)-(E(g(\delta)))^2\\[1ex] &=E\left(\left(\sum_{i=1}^{n-\delta}[X_i=X_{i+\delta}]\right)^2\right)-\left({n-\delta\over 26}\right)^2\\[1ex] &=E\left(\sum_{i}[X_i=X_{i+\delta}]^2+2\sum_{i<j}[X_i=X_{i+\delta}][X_j=X_{j+\delta}]\right)-\left({n-\delta\over 26}\right)^2\\[1ex] &={n-\delta\over 26}+2{(n-\delta)(n-\delta-1)\over 26^2}-\left({n-\delta\over 26}\right)^2\\[1ex] &={25(n-\delta)\over 26^2}\\[1ex] \sigma(g(\delta))&={5\over 26}\sqrt{n-\delta} \end{align}$$
A quick simulation shows that with $n=1000$ (for example), the chart should resemble the following (where the red lines show the above theoretical mean and mean plus/minus three standard deviations):
"Type 2" gap analysis
A different type of "gap analysis" is to focus on each type of ball separately, collecting all the gaps that occur among the draws for each type, then combining all of these to produce an i.i.d. sample of observed gap values. This will produce a histogram corresponding to a Geometric(1/26) distribution, if the draws are i.i.d.Uniform{1,...,26}.
To see this, consider a given occurrence of a 1-ball, and let $Y$ be the the number of draws to get the next occurrence of a 1-ball. Then we have $Y=k$ iff we first get $k-1$ balls that are not a 1-ball, followed by the waited-for 1-ball, i.e.,
$$P(Y=k)=P(\text{not 1-ball})^{k-1}P(\text{1-ball})=\left(1-{1\over 26}\right)^{k-1}\left({1\over 26}\right)\quad (k=1,2,3,...)$$
which is the probability mass function of a Geometric(1/26) distribution.
Here is a quick simulated sample of 1000 i.i.d. Geometric(1/26) RVs, where the histogram is normalized to be comparable to the probability mass function given above, which is plotted by the dots:
As we can see, this is quite like your first plot. Also, note that a goodness-of-fit test might be useful for testing whether a sample departs significantly from the expected distribution.