Ratios of consecutive terms approach Pi alternating from below and above.

Terms

a(0) =1a(1) =3a(2) =10a(3) =31a(4) =98a(5) =307a(6) =965a(7) =3031a(8) =9523a(9) =29917a(10) =93988a(11) =295272a(12) =927625a(13) =2914219a(14) =9155290a(15) =28762191a(16) =90359088a(17) =283871447a(18) =891808453a(19) =2801698884a(20) =8801796632a(21) =27651659637a(22) =86870250776a(23) =272910941653a(24) =857375009382