a(n) = ((4+sqrt(2))*(2+sqrt(2))^n + (4-sqrt(2))*(2-sqrt(2))^n)/4.

Terms

a(0) =2a(1) =5a(2) =16a(3) =54a(4) =184a(5) =628a(6) =2144a(7) =7320a(8) =24992a(9) =85328a(10) =291328a(11) =994656a(12) =3395968a(13) =11594560a(14) =39586304a(15) =135156096a(16) =461451776a(17) =1575494912a(18) =5379076096a(19) =18365314560a(20) =62703106048a(21) =214081795072a(22) =730920968192