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a(n) is the maximal number of congruent n-gons that can be arranged around a vertex without overlapping.
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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