A396817 Total number of separating two-component spanning forests over all unordered pairs of rim vertices in the wheel graph W_n.

24, 156, 770, 3264, 12586, 45528, 157320, 525250, 1707420, 5432832, 16986684, 52341926, 159301560, 479713584, 1431349646, 4236499008, 12450156790, 36357139500, 105569638440, 304977079462, 876968277144, 2511136567296, 7162824312600, 20359351488554, 57680755309656, 162926987593188

Offset

4,1

Comments

Here F_G(u|v) denotes the number of spanning forests of G with exactly two connected components such that u and v belong to different components.

This sequence is defined by a(n) = Sum F_{W_n}(u|v), where the sum is over all unordered pairs {u,v} of rim vertices of the wheel graph W_n.

The wheel graph W_n has n-1 rim vertices.

Links

Table of n, a(n) for n=4..29.

Formula

a(n) = m*((L_{2m}-2)*(F_{2q+1}-1) - F_{2m}*(L_{2q+1}-2q-1)) - e_m*m/2*(F_m*(L_{2m}-2) - F_{2m}*(L_m-2)), where m = n-1, q = floor(m/2), and e_m = (1+(-1)^m)/2; and where F_n is the n-th Fibonacci number and L_n is the n-th Lucas number.

Example

For n=4, the graph W_4 has three unordered pairs of rim vertices. Each pair contributes 8 separating two-component spanning forests, so a(4)=24.

Crossrefs

Cf. A396818, A000045, A000032.

Sequence in context: A004308 A008663 A277984 * A125334 A126492 A222002

Adjacent sequences: A396814 A396815 A396816 * A396818 A396819 A396820

Keyword

nonn,new

Author

Shunya Tamura, Jun 06 2026

Status

approved