W. Lang, Oct 13, 2008

A145375 numerators of the rationals r(n):=sum(((-1)^(k+1))/binomial(2*k,k),k=1..n), n>=1, 
(in lowest terms) for  n=1..25:

[1, 1, 23, 31, 47, 1031, 26827, 134107, 455989, 8663665, 4331849, 187279, 4981622687, 
747243353, 173360460899, 1074834852769, 233659750871, 926581770421, 198844447947463, 
6856705101503, 1630524473145553, 350562761725846217, 97378544923877951, 42247307182355837, 
1922252476797231443]


A145556 denominators of the rationals r(n):=sum(((-1)^(k+1))/binomial(2*k,k),k=1..n), n>=1, 
(in lowest terms) for  n=1..25:

[2, 3, 60, 84, 126, 2772, 72072, 360360, 1225224, 23279256, 11639628, 503217, 13385572200, 
2007835830, 465817912560, 2888071057872, 627841534320, 2489716429200, 534293145706320, 
18423901576080, 4381203794791824, 941958815880242160, 261655226633400600, 113518113708644568, 
5165074173743327844]

Rationals r(n)= A145375(n)/A145556(n), n=1..25: 

[1/2, 1/3, 23/60, 31/84, 47/126, 1031/2772, 26827/72072, 134107/360360, 455989/1225224,
 8663665/23279256, 4331849/11639628, 187279/503217, 4981622687/13385572200, 
747243353/2007835830, 173360460899/465817912560, 1074834852769/2888071057872, 
233659750871/627841534320, 926581770421/2489716429200, 198844447947463/534293145706320, 
6856705101503/18423901576080, 1630524473145553/4381203794791824, 
350562761725846217/941958815880242160, 97378544923877951/261655226633400600, 
42247307182355837/113518113708644568, 1922252476797231443/5165074173743327844,...]


The values r(10^k), for k=0,..3 are (maple11, 10 digits) [.5000000000, .3721624523, .3721635764,
 .3721635764].

This should be compared with the limit n -> infinity which is 
(1 + 4*ln(phi)/(2*phi-1))/5) with the golden section phi:=(1+sqrt(5))/2 which 
is approximatly (maple11, 10 digits)  0.3721635764.


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