seq[n_] := FullSimplify[LerchPhi[1/2, 2, -n] - 2^(-2-n) (π^2 - 6 Log[2]^2)/3];
Array[seq, 10]
(* {1, 3/4, 35/72, 11/36, 347/1800, 149/1200, 9701/117600, 209/3675, 8093/198450, 6031/198450} *)

DifferenceRootReduce[LerchPhi[1/2, 2, -n] - 2^(-2-n) (π^2 - 6 Log[2]^2)/3, n]
(* LerchPhi[1/2, 2, -n] - 2^(-2-n) (π^2 - 6 Log[2]^2)/3 *)