a(n) = Sum_{m=1..n} Sum_{i=1..m} F(i)*F(i+1) where F(n)=Fibonacci numbers A000045.

Terms

a(0) =0a(1) =1a(2) =4a(3) =13a(4) =37a(5) =101a(6) =269a(7) =710a(8) =1865a(9) =4890a(10) =12810a(11) =33546a(12) =87834a(13) =229963a(14) =602062a(15) =1576231a(16) =4126639a(17) =10803695a(18) =28284455a(19) =74049680a(20) =193864595a(21) =507544116a(22) =1328767764a(23) =3478759188a(24) =9107509812a(25) =23843770261a(26) =62423800984a(27) =163427632705