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