a(n) = ((5 + 2*sqrt(2))*(1 + sqrt(2))^n + (5 - 2*sqrt(2))*(1 - sqrt(2))^n)/2.

Terms

a(0) =5a(1) =9a(2) =23a(3) =55a(4) =133a(5) =321a(6) =775a(7) =1871a(8) =4517a(9) =10905a(10) =26327a(11) =63559a(12) =153445a(13) =370449a(14) =894343a(15) =2159135a(16) =5212613a(17) =12584361a(18) =30381335a(19) =73347031a(20) =177075397a(21) =427497825a(22) =1032071047a(23) =2491639919a(24) =6015350885a(25) =14522341689a(26) =35060034263a(27) =84642410215