W. Lang:

More information on A096954 (From Machin's formula)

The rational numbers M(n):=A096954(n)/A096955(n) are, for n=0..10:

[951/1195, 1339849258/1706489875, 9569810428334921/12184551018734375, 

19132121777295048135244/24359780855939418203125, 

81963468350564671450762204559/104359128170408663038552734375, 

1287504688596138051498743351405666674/
1639301884061026141391921953564453125, 

23901655485793371607250742363386659018053931/
30432532948821209122295591520605416259765625, 

170660807873601670198453967268421248219727522686104/
217292089321202035784330810406062747771759033203125, 

828606860556905456887031070939684222209660202633295159013/
1055015021899892426313124618802400368315005057525634765625, 

22482154928888407431983565181298508267079237911238292824980324914/
28625168706323283759630195891540657933297666848187847137451171875, 

32105079292325884351865900505769360289444947149921276835678985319103709/
40877456541847307290845910488017348045197400700883450408458709716796875]


and their values are (mapl9 20 digits)

 [.79581589958158995816, .78514925733151507858, .78540525733125860626, 

 .78539794304554432375, 

 .78539817060109987931, 

 .78539816315382715204, 
  
 .78539816340588869050, 

 .78539816339715055717, 

 .78539816339745896187, 

 .78539816339744792423, 
 
 .78539816339744832369]


They should be compared with the value of Pi/4 which is (maple9, 20 digits)

.78539816339744830962

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The numerators of the rationals M(n) (in lowest terms) are, for n=0..15: 

[951, 1339849258, 9569810428334921, 19132121777295048135244, 

81963468350564671450762204559, 1287504688596138051498743351405666674, 

23901655485793371607250742363386659018053931, 

170660807873601670198453967268421248219727522686104, 

828606860556905456887031070939684222209660202633295159013, 

22482154928888407431983565181298508267079237911238292824980324914, 

32105079292325884351865900505769360289444947149921276835678985319103709, 

1054477684697744413449111797973540542314209132163915500121291245623612265689956, 

37645512392262411676124861935161956688822510560984123522435763797997989319666144061477, 

6451047940075263651892462035687220645593848730893809576579986166346072398703739771226174238, 

267155474304153372918570147036737083422401815564924050394148118252401532301378633919427673916629321, 

11826645581988847231222857907505771480517030826468625401668689633538572570099880238779327838251428935708336]


and the corresponding denominators are:


[1195, 1706489875, 12184551018734375, 24359780855939418203125, 

104359128170408663038552734375, 1639301884061026141391921953564453125, 

30432532948821209122295591520605416259765625, 

217292089321202035784330810406062747771759033203125, 

1055015021899892426313124618802400368315005057525634765625, 

28625168706323283759630195891540657933297666848187847137451171875, 

40877456541847307290845910488017348045197400700883450408458709716796875, 

1342602687197944522862235320466972389171589417125219053199402725696563720703125, 

47931755059646118181508589837746206151168974434756023461126926934070885181427001953125, 

8213729342286137749937856480365703124682768966063096448369093580203189097344875335693359375, 

340152914476526693950295217510852762934128223430105345864785970936220114211341552436351776123046875, 

15058152836555602995979480167564001020459262144177061782385390570532067586496181632955558598041534423828125]


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