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

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, 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, 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

Offset: 0

Author

Kival Ngaokrajang, Jan 02 2016

Keywords

• nonn
• tabf

Comments

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.

T(n,k) is conjectured to be even rows of A109004 (excluding the first column).

Examples

			Irregular triangle begins:
n\k   0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
0     1  2
1     1  2  1  4
2     1  2  3  2  1  6
3     1  2  1  4  1  2  1  8
4     1  2  1  2  5  2  1  2  1 10
5     1  2  3  4  1  6  1  4  3  2  1 12
6     1  2  1  2  1  2  7  2  1  2  1  2  1 14
7     1  2  1  4  1  2  1  8  1  2  1  4  1  2  1 16
...
		

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A109004, A262343, A264906.

Programs

Mathematica

Table[GCD[2 n + 3 + k, k + 1], {n, 0, 8}, {k, 0, 2 n + 1}] // Flatten (* Michael De Vlieger, Jan 03 2016 *)

PARI

for (n=0, 20,for (k=0, 2*n+2, t=gcd(2*n+3+k, k+1); print1(t, ", ")))

Formula

T(n,k) = gcd(2*n+3+k, k+1), n >= 0, k = 0..2*n+1.