Rationals r(n) = A130413(n)/A130414(n)

 r(n):= 1+ (4/3)*sum(((-1)^(j+1))/((2*j+1)*((2*j+1)^2-1)),j=1..n), n>=0, 

 obtained from the series

 3/4 + sum(((-1)^(j+1))/((2*j+1)*((2*j+1)^2-1)),j=1..n) with limit Pi/4,
 
 attributed to K. G. Nilakantha (c. 1500). See the Ranjan Roy reference given  

 in A130411.


r(n), n=0..25:

[1, 19/18, 47/45, 1321/1260, 989/945, 21779/20790, 141481/135135, 1132277/1081080, 801821/765765, 91424611/87297210, 45706007/43648605, 4205393539/4015671660, 5256312899/5019589575, 31539920369/30117537450, 457304942543/436704293025, 226832956041173/216605329340400, 14176557010703/13537833083775, 28353956712541/27075666167550, 524535004412921/500899824099675, 2098185082863029/2003599296398700, 21505999559803111/20536892788086675, 1849545596908995521/1766172779775454050, 924759870824723383/883086389887727025, 347713977547921536583/332040482597785361400, 304246436302024620449/290535422273062191225, 608498715448137185839/581070844546124382450]


Numerators A130413, n=0..25:

[1,19,47,1321,989,21779,141481,1132277,801821,91424611,45706007,4205393539,5256312899,31539920369,457304942543,226832956041173,14176557010703,28353956712541,524535004412921,2098185082863029,21505999559803111,1849545596908995521,924759870824723383,347713977547921536583,304246436302024620449,608498715448137185839]


Denominators A130414, n=0..25:

[1, 18, 45, 1260, 945, 20790, 135135, 1081080, 765765, 87297210, 43648605, 4015671660, 5019589575, 30117537450, 436704293025, 216605329340400, 13537833083775, 27075666167550, 500899824099675, 2003599296398700, 20536892788086675, 1766172779775454050, 883086389887727025, 332040482597785361400, 290535422273062191225, 581070844546124382450]


The values for r(10^k), k=0,1,2 are

 [1.055555556, 1.047135573, 1.047197470].

They should be compared with the value for Pi/3 (maple10, 10 digits)

 1.047197551 


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