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%I #12 Jun 16 2026 19:06:25
%S 24,156,770,3264,12586,45528,157320,525250,1707420,5432832,16986684,
%T 52341926,159301560,479713584,1431349646,4236499008,12450156790,
%U 36357139500,105569638440,304977079462,876968277144,2511136567296,7162824312600,20359351488554,57680755309656,162926987593188
%N Total number of separating two-component spanning forests over all unordered pairs of rim vertices in the wheel graph W_n.
%C 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.
%C 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.
%C The wheel graph W_n has n-1 rim vertices.
%F 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.
%e 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.
%Y Cf. A396818, A000045, A000032.
%K nonn
%O 4,1
%A _Shunya Tamura_, Jun 06 2026