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A337301 Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.

%I A337301 #48 Oct 21 2020 23:21:36
%S A337301 1,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,3,3,3,3,2,2,3,3,3,3,3,2,2,3,3,4,4,3,
%T A337301 3,2,2,3,3,4,4,4,3,3,2,2,3,3,4,4,4,4,3,3,2,2,3,4,4,4,4,4,4,4,3,2,2,3,
%U A337301 4,4,5,5,5,5,4,4,3,2,2,3,4,4,5,5,5,5,5,4,4,3,2
%N A337301 Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.
%F A337301 T(n,k) = round(sin((k+1)*Pi/n)/sin(Pi/n)), n >= 4, 1 <= k <= n-3.
%e A337301 Triangle begins:
%e A337301 1;
%e A337301 2, 2;
%e A337301 2, 2, 2;
%e A337301 2, 2, 2, 2;
%e A337301 2, 2, 3, 2, 2;
%e A337301 2, 3, 3, 3, 3, 2;
%e A337301 2, 3, 3, 3, 3, 3, 2;
%e A337301 2, 3, 3, 4, 4, 3, 3, 2;
%e A337301 2, 3, 3, 4, 4, 4, 3, 3, 2;
%e A337301 2, 3, 3, 4, 4, 4, 4, 3, 3, 2;
%e A337301 2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2;
%e A337301 2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2;
%e A337301 2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2;
%e A337301 ...
%e A337301 Row n lists the closest integers to the length of the diagonals drawn from a fixed vertex of a regular n-gon with unit edge length, n >= 4.
%e A337301 The lengths of the diagonals drawn from vertex A of a regular 8-gon ABCDEFGH with unit edge length are:
%e A337301 AC = 1.84775...
%e A337301 AD = 2.41421...
%e A337301 AE = 2.61312...
%e A337301 AF = 2.41421...
%e A337301 AG = 1.84775...
%e A337301 So the row for n=8 is 2, 2, 3, 2, 2.
%t A337301 T[n_,k_]:=Round[Sin[(k+1)*Pi/n]/Sin[Pi/n]]; Flatten[Table[T[n,k],{n,4,16},{k,1,n-3}]] (* _Stefano Spezia_, Sep 07 2020 *)
%Y A337301 Cf. A064313.
%Y A337301 Decimal expansion of diagonal lengths of regular n-gons with unit edge length:
%Y A337301 n=4  A002193.
%Y A337301 n=5  A001622.
%Y A337301 n=6  A002194, A000038.
%Y A337301 n=7  A160389, A231187.
%Y A337301 n=8  A179260, A014176, A121601.
%Y A337301 n=9  A332437.
%Y A337301 n=10 A188593, A104457, A019970, A134945.
%Y A337301 n=11 A231186.
%Y A337301 n=12 A188887, A090388, A337402, A019973, A214726.
%K A337301 easy,nonn,tabl
%O A337301 4,2
%A A337301 _Mohammed Yaseen_, Aug 22 2020