A396670 Binary colorings of the sunlet graph S_n up to rotation, with no monochromatic connected component of size 3 or more.

4, 11, 16, 38, 68, 151, 312, 684, 1476, 3293, 7280, 16358, 36788, 83311, 189120, 431392, 986244, 2262307, 5200848, 11985862, 27676612, 64034953, 148406240, 344507804, 800902876, 1864502925, 4346071600, 10142619038, 23696518916, 55420734751, 129742923472

Offset

3,1

Comments

The sunlet graph S_n is obtained from the cycle C_n by attaching one pendant vertex to each cycle vertex. Thus S_n has 2n vertices.

This is the rotation-only version. Reflected colorings are not identified.

Equivalently, runs of equal colors around the cycle have length at most 2. A pendant vertex attached to a cycle vertex in a run of length 2 is forced to have the opposite color, a pendant vertex attached to an isolated cycle vertex is free.

Links

Table of n, a(n) for n=3..33.

Robert P. P. McKone, The rotation-class representative Sunlets a(3) to a(11).

Robert P. P. McKone, a(3) = 4 Sunlet Graphs.

Robert P. P. McKone, a(4) = 11 Sunlet Graphs.

Robert P. P. McKone, a(5) = 16 Sunlet Graphs

Robert P. P. McKone, a(6) = 38 Sunlet Graphs.

Robert P. P. McKone, a(7) = 68 Sunlet Graphs.

Robert P. P. McKone, a(8) = 151 Sunlet Graphs.

Robert P. P. McKone, a(9) = 312 Sunlet Graphs.

Robert P. P. McKone, a(10) = 684 Sunlet Graphs.

Robert P. P. McKone, a(11) = 1476 Sunlet Graphs.

Eric Weisstein's World of Mathematics, Sunlet Graph

Formula

a(n) = (1/n) * Sum_{d|n} (phi(n/d) * ((1+sqrt(2))^d + (1-sqrt(2))^d + 2*(-1)^d)).

Example

Binary sunlet colorings up to rotation.  No monochromatic component has size 3 or more.  Blocks are rl around the cycle, ring color, then pendant color.
The a(3) = 4 rotation-class representative sunlets, {00,10,10}, {01,01,10}, {01,10,10}, {01,01,11}

Mathematica

a[n_] := Simplify[Sum[EulerPhi[n/d] ((1 + Sqrt[2])^d + (1 - Sqrt[2])^d + 2 (-1)^d), {d, Divisors[n]}]/n];
Table[a[n], {n, 3, 33}]

Crossrefs

Cf. A001868 (the total number of binary Sunlet colorings up to rotation before applying the restriction).

Sequence in context: A054870 A197567 A207460 * A107988 A038241 A360134

Adjacent sequences: A396667 A396668 A396669 * A396671 A396672 A396673

Keyword

nonn

Author

Robert P. P. McKone, Jun 02 2026

Status

approved