a(n) = Sum_{k=0..n} F(k+1)^2*F(n-k+1)^2 where F = Fibonacci numbers (A000045).

Terms

a(0) =1a(1) =2a(2) =9a(3) =26a(4) =84a(5) =250a(6) =747a(7) =2182a(8) =6323a(9) =18132a(10) =51624a(11) =146004a(12) =410677a(13) =1149578a(14) =3204477a(15) =8899502a(16) =24634620a(17) =67990414a(18) =187154271a(19) =513939214a(20) =1408246247a(21) =3851081256a(22) =10512259920a(23) =28647203880a(24) =77946605545a(25) =211782868754