Rationals r(n)=  A120088(n)/A120777(n) defined by

r(n):=1 + (sum(((-1)^k)*C(k)/4^k))/2,k=0..n) with the Catalan numbers C(n)=A000108(n).

 The first instances are: (n=0..30))

[3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024, 2911/2048, 46147/32768, 93009/65536, 369605/262144, 
743409/524288, 5917879/4194304, 11887761/8388608, 47365319/33554432, 95064943/67108864, 
3032383331/2147483648, 6082445497/4294967296, 24264959593/17179869184, 48649328861/34359738368, 
388310999293/274877906944, 778263028691/549755813888, 3106935548009/2199023255552, 
6225306416473/4398046511104, 99433372856743/70368744177664, 199189221750317/140737488355328, 
795541400400905/562949953421312, 1593378719935829/1125899906842624, 12729646371757631/9007199254740992, 
25492261237484057/18014398509481984, 101843764672854807/72057594037927936, 203925966002090383/144115188075855872]
 
 The values (maple10 10 digits are, for n=10^k, k=0..3:

 1.375000000, 1.417940140, 1.414352001, 1.414218014]
 
 This should be compared with the value for sqrt(2) with 10 digits
 
 1.414213562

 

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