Number of Regions by dividing Circle by Chords

Theorem

Let $n$ points be marked on the circumference of a circle $C$.

Let chords be drawn between each pair of these points.

For each $n$, the maximum number $C \left({n}\right)$ of regions into which $C$ can be divided is as follows:

$n$	$C \left({n}\right)$
$1$	$1$
$2$	$2$
$3$	$4$
$4$	$8$
$5$	$16$
$6$	$31$
$7$	$57$
$8$	$99$
$9$	$163$
$10$	$256$

This sequence is A000127 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

This theorem requires a proof. In particular: There is an equation which defines this -- I think it can be found in Graham, Knuth & Patashnik but that's just too far for me to reach at the moment without me getting out of this chair. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Proof

$\quad$ $\quad$

$\quad$ $\quad$

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $31$