# I noticed a pattern, does this have a name?

First of all I am a programmer, not a mathematician, so I may articulate what I am trying to say very poorly. I was working with powers of $2$ when I noticed a relationship I had never noticed before. Basically, if you look at the powers of $2$, every other power of two can be added together make a multiple of 10. Furthermore, if the two numbers are $2^{10}$ and $2^8$, adding them together will result in $2^7\cdot 10$. Put another way, $2^n + 2^{n-2} = 2^{n-3}\cdot 10$.

Does this pattern have a name? I am attaching an image just to clarify my poor explanation. Thanks all! $$\begin{matrix} & & \vdash & 20 & \dashv & \vdash & 160 & \dashv & \vdash & 1280 & \dashv \\ 1 & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512 & 1024 \\ & \vdash & 10 & \dashv & \vdash & 80 & \dashv & \vdash & 640 & \dashv \\ \end{matrix}$$

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I doubt it has a name because it's a fairly simple consequence of the definitions: $2^n+2^{n-2}=2^{n-3}(2^3+2^1)=2^{n-3}\times 10$. It's still a pretty cool pattern though! (And actually demonstrating that nobody has given it a name, if that is in fact the case, is likely to be quite difficult!) –  Matt Pressland Apr 10 '12 at 15:48
This picture needs a LaTeX-wiz. \rlap, \underbrace and \phantom... and all that. –  user2468 Apr 10 '12 at 16:02
I TeXified the picture. Please make sure I did not change it. –  user2468 Apr 10 '12 at 16:11
It seems to be the same! –  Brendon Dugan Apr 10 '12 at 16:17

$$2^n+2^{n+2}=2^n(1+2^2)=2^n\cdot 5 = 10\cdot 2^{n-1}$$
For $n=8$ we have that indeed $2^n+2^{n+2}=2^8+2^{10}=2^7\cdot10$ as you have mentioned.