A396818 - OEIS

%I #12 Jun 16 2026 19:00:01

%S 48,240,1045,4128,15225,53424,180576,592900,1902241,5989248,18564793,

%T 56791280,171782160,514566528,1528298725,4505245344,13192832001,

%U 38403822600,111195838656,320408071588,919203450913,2626517214720,7477481038225,21216084771128,60009979734000,169250827737264

%N Total number of separating two-component spanning forests over all unordered pairs of 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 vertices of the wheel graph W_n.

%C Equivalently, a(n) = b(n) + (n-1)*F_{2n-2}, where b(n) is the corresponding sum over all unordered pairs of rim vertices, F_n is the n-th Fibonacci number, and 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)) + m*F_{2m}, 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 four vertices. The sum over all unordered pairs of rim vertices is 24, and the three center-rim pairs each contribute F_6=8. Hence a(4)=24+3*8=48.

%Y Cf. A396817, A000045, A000032.

%K nonn

%O 4,1

%A _Shunya Tamura_, Jun 06 2026