Number of compositions of n whose run-lengths cover an initial interval of positive integers.

Terms

a(0) =1a(1) =1a(2) =1a(3) =3a(4) =6a(5) =13a(6) =21a(7) =48a(8) =89a(9) =180a(10) =355a(11) =707a(12) =1382a(13) =2758a(14) =5448a(15) =10786a(16) =21391a(17) =42476a(18) =84291a(19) =167516a(20) =333036a(21) =662153a(22) =1317687a(23) =2622706a(24) =5221951a(25) =10400350a(26) =20720877a(27) =41288823a(28) =82294979a(29) =164052035