Solution (a(n)) of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-2); see Comments.

Terms

a(0) =1a(1) =2a(2) =6a(3) =14a(4) =27a(5) =47a(6) =75a(7) =112a(8) =159a(9) =217a(10) =287a(11) =370a(12) =468a(13) =582a(14) =713a(15) =862a(16) =1030a(17) =1218a(18) =1427a(19) =1658a(20) =1912a(21) =2190a(22) =2493a(23) =2822a(24) =3179a(25) =3565a(26) =3981a(27) =4428a(28) =4907a(29) =5419