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A266685 T(n,k) is the number of loops appearing in pattern of circular arc connecting two vertices of regular polygons. (See Comments.)

%I A266685 #13 Mar 22 2017 10:34:46
%S A266685 1,2,1,1,2,1,4,1,1,2,3,2,1,6,1,1,2,1,4,1,2,1,8,1,1,2,1,2,5,2,1,2,1,10,
%T A266685 1,1,2,3,4,1,6,1,4,3,2,1,12,1,1,2,1,2,1,2,7,2,1,2,1,2,1,14,1,1,2,1,4,
%U A266685 1,2,1,8,1,2,1,4,1,2,1,16,1,1,2,3,2,1,6,1,2,9,2,1,6,1,2,3,2,1,18
%N A266685 T(n,k) is the number of loops appearing in pattern of circular arc connecting two vertices of regular polygons. (See Comments.)
%C A266685 The patterns in A262343 and A264906 can be considered as case of skip 0 and 1 vertex of circle construction on regular polygons. k is the cyclic number of loops of the case skip n-vertices. See illustration for more details.
%C A266685 T(n,k) is conjectured to be even rows of A109004 (excluding the first column).
%H A266685 Kival Ngaokrajang, <a href="/A266685/a266685.pdf">Illustration of initial terms</a>
%F A266685 T(n,k) = gcd(2*n+3+k, k+1), n >= 0, k = 0..2*n+1.
%e A266685 Irregular triangle begins:
%e A266685 n\k   0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
%e A266685 0     1  2
%e A266685 1     1  2  1  4
%e A266685 2     1  2  3  2  1  6
%e A266685 3     1  2  1  4  1  2  1  8
%e A266685 4     1  2  1  2  5  2  1  2  1 10
%e A266685 5     1  2  3  4  1  6  1  4  3  2  1 12
%e A266685 6     1  2  1  2  1  2  7  2  1  2  1  2  1 14
%e A266685 7     1  2  1  4  1  2  1  8  1  2  1  4  1  2  1 16
%e A266685 ...
%t A266685 Table[GCD[2 n + 3 + k, k + 1], {n, 0, 8}, {k, 0, 2 n + 1}] // Flatten (* _Michael De Vlieger_, Jan 03 2016 *)
%o A266685 (PARI) for (n=0, 20,for (k=0, 2*n+2, t=gcd(2*n+3+k, k+1); print1(t, ", ")))
%Y A266685 Cf. A109004, A262343, A264906.
%K A266685 nonn,tabf
%O A266685 0,2
%A A266685 _Kival Ngaokrajang_, Jan 02 2016