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%I #18 Jul 17 2024 09:42:51
%S 6,4,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N a(n) is the maximal number of congruent n-gons that can be arranged around a vertex without overlapping.
%C As n increases, the internal angle of the n-gon tends towards 180 degrees, so a(n) = 2 for n > 6.
%C This also shows that no regular n-gon can tile the plane for n > 6 since in any tiling by convex tiles at least three tiles meet at every vertex.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F a(n) = floor(2*n/(n-2)).
%e For n = 5: Arranging 3 pentagons around a vertex leaves a gap smaller than the internal angle of a pentagon, so a(5) = 3.
%t Table[Floor[2 n/(n - 2)], {n, 3, 100}] (* _Wesley Ivan Hurt_, Apr 19 2021 *)
%o (PARI) a(n) = floor(n*(2/(n-2)))
%o (Magma) [Floor(2*n/(n-2)) : n in [3..100]]; // _Wesley Ivan Hurt_, Apr 19 2021
%Y Cf. A071279.
%K nonn,easy
%O 3,1
%A _Felix Fröhlich_, Apr 16 2021