Rationals r(n)= A121503(n)/(4*A120785(n)) r(n):= 4-(sum(C(k)*(1+2^(k+1))/16^k,k=0..n)/4, n>=0, with C(k)=A000108(k) (Catalan numbers). Partial sums of a series with value sqrt(2) and sqrt(3). r(n), n=0..30: [13/4, 203/64, 1615/512, 51595/16384, 412529/131072, 6599099/2097152, 52788535/16777216, 3378355987/1073741824, 27026481101/8589934592, 432421205841/137438953472, 3459361042977/1099511627776, 110699432952143/35184372088832, 885595037556565/281474976710656, 14169517557800915/4503599627370496, 113356129507566775/36028797018963968, 14509583941597490435/4611686018427387904, 116076669215561983525/36893488147419103232, 1857226690456124999125/590295810358705651712, 14857813461043820896925/4722366482869645213696, 475450029826846501711285/151115727451828646838272, 3803600235173280805899215/1208925819614629174706176, 60857603737117746371057165/19342813113834066795298816, 486860829801015538881760225/154742504910672534362390528, 31159093104387201697361548375/9903520314283042199192993792, 249272744824277113006941065177/79228162514264337593543950336, 3988363917106863881937987524837/1267650600228229401496703205376, 31906911336546758003368223375845/10141204801825835211973625643008, 1021021162764829938478540572438583/324518553658426726783156020576256, 8168169302100939682371062614348829/2596148429267413814265248164610048, 130690708833480516244587091171447939/41538374868278621028243970633760768, 1045525670667332091135796796426037431/332306998946228968225951765070086144] Numerators are A121503(n). For n=0..30: [13, 203, 1615, 51595, 412529, 6599099, 52788535, 3378355987, 27026481101, 432421205841, 3459361042977, 110699432952143, 885595037556565, 14169517557800915, 113356129507566775, 14509583941597490435, 116076669215561983525, 1857226690456124999125, 14857813461043820896925, 475450029826846501711285, 3803600235173280805899215, 60857603737117746371057165, 486860829801015538881760225, 31159093104387201697361548375, 249272744824277113006941065177, 3988363917106863881937987524837, 31906911336546758003368223375845, 1021021162764829938478540572438583, 8168169302100939682371062614348829, 130690708833480516244587091171447939, 1045525670667332091135796796426037431] The denominators are, for n=0..30: [4, 64, 512, 16384, 131072, 2097152, 16777216, 1073741824, 8589934592, 137438953472, 1099511627776, 35184372088832, 281474976710656, 4503599627370496, 36028797018963968, 4611686018427387904, 36893488147419103232, 590295810358705651712, 4722366482869645213696, 151115727451828646838272, 1208925819614629174706176, 19342813113834066795298816, 154742504910672534362390528, 9903520314283042199192993792, 79228162514264337593543950336, 1267650600228229401496703205376, 10141204801825835211973625643008, 324518553658426726783156020576256, 2596148429267413814265248164610048, 41538374868278621028243970633760768, 332306998946228968225951765070086144] The dominators divided by 4 are A120785(n), n=0..30: [1, 16, 128, 4096, 32768, 524288, 4194304, 268435456, 2147483648, 34359738368, 274877906944, 8796093022208, 70368744177664, 1125899906842624, 9007199254740992, 1152921504606846976, 9223372036854775808, 147573952589676412928, 1180591620717411303424, 37778931862957161709568, 302231454903657293676544, 4835703278458516698824704, 38685626227668133590597632, 2475880078570760549798248448, 19807040628566084398385987584, 316912650057057350374175801344, 2535301200456458802993406410752, 81129638414606681695789005144064, 649037107316853453566312041152512, 10384593717069655257060992658440192, 83076749736557242056487941267521536] ######################################################################################################################### Some numerical values for r(n) are, for n=10^k, k=0..3: [3.171875000, 3.146270540, 3.146264370, 3.146264370] (maple10, 10 digits). This should be compared with the limit sqrt(2)+sqrt(3) which is 3.146264369 (maple10, 10 digits). ######################################################################################################################## Popper (see the reference) notes that (sqrt(2)+sqrt(3)-pi)/pi = .0015 (rounded). sqrt(2) + sqrt(3) = (4*sin(Pi/4) + 6*tan(Pi/6))/2. This is the artihmetic mean of the areas of a regular 8-gon, resp. a regular 6-gon inscribing, resp. circumscribing a unit circle which has area Pi. ############################################ e.o.f. ####################################################################