[seqfan] Re: A series for exp(Pi)

[Jaume Oliver i Lafont]

Thanks again, -Alexander, Richard, Maximilian-, for your work.

with x=1/2, exp(6asin(x)) computes exp(Pi),
and with x=-1/2, it computes its reciprocal 1/exp(Pi)

exp(Pi) as computed by plugging 1/2, one more term each line.
a(n)=if(n<2,[1,6][n+1],((n-2)^2+36)*a(n-2))
b(n)=if(!n,1,-2*n*b(n-1))
s=0; for (n=0,25,s+=a(n)/b(n);print(s*1.))

1.000000000000000000000000000
4.000000000000000000000000000
8.500000000000000000000000000
13.12500000000000000000000000
16.87500000000000000000000000
19.47656250000000000000000000
21.10156250000000000000000000
22.04617745535714285714285714
22.56849888392857142857142857
22.84729149228050595238095238
22.99238077799479166666666667
23.06651426703382880140692641
23.10388574971781148538961039
23.12253792564590255935470779
23.13177812740842575044932746
23.13633014653373369111938099
23.13856319529301012896724741
23.13965517782215109635225106
23.14018789860459286093687279
23.14044732427854813462117556
23.14057349499017907886490176
23.14063479974765541437244235
23.14066456729650340771133013
23.14067901510743038994309736
23.14068602558089096808450209
23.14068942683638002848489729

exp(Pi) as computed by plugging -1/2 and then inverting the result
// a(n) the same as before
b(n)=if(!n,1,-2*n*b(n-1))
s=0; for (n=0,25,s+=a(n)/b(n);print(1./s))
1.000000000000000000000000000
-0.5000000000000000000000000000
0.4000000000000000000000000000
-0.4705882352941176470588235294
0.6153846153846153846153846154
-1.024000000000000000000000000
1.542168674698795180722891566
-3.376354215732454074422986340
4.421961752004935225169648365
-18.99384471003891909796019763
10.81775164672776721007372939
54.62343236258507389984620172
17.96020743645145456136266233
27.00770681323076388733446709
21.61383126966584175758151264
23.97239285691561645574523350
22.75431949918253546357694280
23.33410957158627851806014370
23.04761469650443776683304108
23.18624859190561371987319442
23.11861680114008971589106302
23.15142888596491787456660092
23.13548480540554593992088942
23.14322059144052715847967643
23.13946633025879637420577321
23.14128762450017775606900208

Maybe some of these approximations helps give some clue why exp(Pi)-Pi
is almost integer. Are there two rational approximations for exp(Pi)
and Pi whose difference is 20?