a(n) = Sum_{k>=1} floor( 2*(1+sqrt(2))^(n-k) ).

Terms

a(0) =2a(1) =6a(2) =17a(3) =45a(4) =112a(5) =276a(6) =671a(7) =1627a(8) =3934a(9) =9506a(10) =22957a(11) =55433a(12) =133836a(13) =323120a(14) =780091a(15) =1883319a(16) =4546746a(17) =10976830a(18) =26500425a(19) =63977701a(20) =154455848a(21) =372889420a(22) =900234711a(23) =2173358867a(24) =5246952470a(25) =12667263834