Wikipedia:Reference desk/Archives/Mathematics/2020 January 11

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January 11

Number of regions in regular polygon with diagonals drawn

Hi. At https://oeis.org/A007678 there is a Mathematica formula that is said to generate the stated sequence, but I am curious to know whether this really does work for any n, however large. As you can see, the formula incorporates a number of special-case divisibility tests, but these stop at 210. Intuitively I would think these special cases would have to go on indefinitely for larger and larger n, but I could be wrong. Note that the adjacent "PARI" formula works only for odd n (which is the "easy" case), although it does not actually say this. This doesn't give me full confidence that any restrictions on the Mathematica formula would necessarily be mentioned either. — Preceding unsigned comment added by 2A00:23C5:4B91:AB00:995C:88E4:F919:7EE4 (talk) 22:00, 11 January 2020 (UTC)

See this paper of Poonen and Rubinstein. --JBL (talk) 23:59, 11 January 2020 (UTC)

Thanks for the link. I'm glad that I did not try to work this out myself. — Preceding unsigned comment added by 2A00:23C5:4B91:AB00:995C:88E4:F919:7EE4 (talk) 01:25, 12 January 2020 (UTC)

See Langley’s Adventitious Angles#Generalization for a bit more on this. Turns out, though it's not obvious why, that this problem is equivalent to the problem of finding adventitious quadrilaterals. --RDBury (talk) 09:01, 13 January 2020 (UTC)