Solution (a(n)) of the complementary equation in Comments.

Terms

a(0) =1a(1) =6a(2) =19a(3) =45a(4) =102a(5) =215a(6) =445a(7) =904a(8) =1826a(9) =3669a(10) =7359a(11) =14738a(12) =29500a(13) =59023a(14) =118073a(15) =236172a(16) =472376a(17) =944782a(18) =1889599a(19) =3779231a(20) =7558500a(21) =15117036a(22) =30234113a(23) =60468265a(24) =120936574a(25) =241873190a(26) =483746427a(27) =967492899a(28) =1934985848a(29) =3869971744