a(n) = (F(2n+1) + F(2n-1) + F(n+3) - 2)/2, where F() = Fibonacci numbers A000045.

Terms

a(0) =2a(1) =5a(2) =12a(3) =29a(4) =71a(5) =177a(6) =448a(7) =1147a(8) =2960a(9) =7679a(10) =19989a(11) =52145a(12) =136214a(13) =356121a(14) =931540a(15) =2437513a(16) =6379403a(17) =16698113a(18) =43710756a(19) =114427391a(20) =299560472a(21) =784236315a(22) =2053119817a(23) =5375076769a(24) =14072035466a(25) =36840908237a(26) =96450492828a(27) =252510252437