A266685 - OEIS

%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