a(n) = Sum_{k=floor((n+1)/2)..n} J(k+1), J(k) = A001045(k).

Terms

a(0) =1a(1) =1a(2) =4a(3) =8a(4) =19a(5) =37a(6) =80a(7) =160a(8) =331a(9) =661a(10) =1344a(11) =2688a(12) =5419a(13) =10837a(14) =21760a(15) =43520a(16) =87211a(17) =174421a(18) =349184a(19) =698368a(20) =1397419a(21) =2794837a(22) =5591040a(23) =11182080a(24) =22366891a(25) =44733781a(26) =89473024a(27) =178946048a(28) =357903019a(29) =715806037