In Photoshop and many other graphic software you have layer blending modes Add and Subtract. With subtract you can calculate the step size (=B-A) between two colors A and B. You can add that step to B to get as distant new color C or subtract it from A to see what belongs in front of A to that series.
A nice theory, but it works poorly in normal RGB mode with 0 to 255 channel value range. We have unfortunately no negative colors. You get better results in 32 bit RGB mode. See an example:
I have taken 5th and 6th colors of your series and calculated the 7th. For reference the original 7th color is shown as the right half of the image, the calculated (=extrapolated) version is the left half.
Another example: I have subtracted and merged 2nd and 6th color of your series. Then I subtract it with 50% opacity from your 2nd color. I get your first color. You see that the first color in the reference series has nearly sunken to the calculated color. :
Actually you do not need graphic software. You know the RGB values and you can calculate the colors with a calculator or spreadsheet. It's elementary, if you use linear interpolation. Calculate X1 plus or minus N*(X2-X1)/M where X is R, G and B in their turn, N is an integer "how many steps you want to advance", M is another integer "how many steps there already is from color1 to color2, use plus and minus for different directions. Do not make any roundings until you are ready. If you happen to get a final R,G or B number below zero or over 255, then you have wanted something impossible in 8 bit RGB range.
NOTE: This is not exact even with 32 bit colors because we used elementary linear interpolation with no idea how the series was actually formed. A series between two colors can be formed via arbitary routes in the color space. Linear interpolation is only one possiblity and it happened to give quite the same result than your color series generator just in this case. Another source for inaccuracy is that I used as starting values numbers which are already rounded to 8 bit. Finally the system color management and the trip via Imgur distort colors.
linear interpolation between RGB numbers is only one of the zillion possiblities to advance from color 1 to color 2 and create some intermediate steps. Somehow visually optimal result needs surely something advanced which takes into the account the limitations of the devices and the human color perception. Artists do all that intuitively.
If you want to get a series without radically differently saturated intermediate colors, you can try interpolating between hue, saturation and luminance values. Note that you must select do you want to go clockwise or CCW along the hue circle.