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User:Jaume Oliver Lafont/Pi

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(2 Around Leibniz-Gregory-Madhava series http://numbers.computation.free.fr/Constants/Pi/piSeries.html)

may be added as a comment to sequences

0,1,-1 period 3

(9*n^2+9*n+2)/2

1,2,4,5,7,8,10,11, (not divisible by 3)


The same constant -not on the OEIS, but see A086089- appears in a faster series:

http://mathworld.wolfram.com/PiFormulas.html (27) (Gosper).


(A093954)

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This formula appears as a comment to A007310 and is equivalent to (33) in http://mathworld.wolfram.com/PiFormulas.html.

A similar one can be written for .


(how can this be checked with a link to WolframAlpha?)


Similarly,

leads to (PARI)

a(n)=((n+1)!*sum(k=0,n,[1,0,1,0,-1,0,-1,0][k%8+1]/(k+1)))

1, 2, 8, 32, 136, 816, 4992, 39936, 399744, 3997440, 47600640, 571207680,

and

b(n)=((n+1)!*sum(k=0,n,[1,0,-1,0][k%4+1]/(k+1)))

1, 2, 4, 16, 104, 624, 3648, 29184, 302976, 3029760, 29698560, 356382720,

with approaching (slowly) as grows.

Other sequences with the same property can be defined, given the zeros at even positions in both sequences, as follows:

a(n)=((2*(n+1))!/(n+1)!/2^(n+1)*sum(k=0,n,[1,1,-1,-1][k%4+1]/(2*k+1)))

1, 4, 17, 104, 1041, 12396, 150753, 2126160,

(closely related to A127676 and first four terms equal)

and

b(n)=((2*(n+1))!/(n+1)!/2^(n+1)*sum(k=0,n,[1,-1][k%2+1]/(2*k+1)))

1, 2, 13, 76, 789, 7734, 110937, 1528920,

(a shifted version of A024199)


For faster convergence, try the same with pairs ((13),(15)) -for - and ((13),(16)) -for - taken from http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf.


From (15)

a(n)=(n+1)!*sum(k=0,n,[2,0,1,0,2,0,-2,0,-1,0,-2,0][k%12+1]/(k+1)/2^(k/2))

2, 4, 13, 52, 272, 1632, 11244, 89952, 807048, 8070480, 88548480, 1062581760,

with a(n)/(n+1)! -> Pi*sqrt(2)/2


From (13)

b(n)=(n+1)!*sum(k=0,n,sin((k+1)*Pi/4)/(k+1)/2^((k-1)/2))

1, 3, 10, 40, 194, 1134, 7848, 62784, 567576, 5698440, 62796240, 753554880,

with b(n)/(n+1)! -> Pi/2


Combining both,

a(n)/b(n) -> sqrt(2)

[ b(k+1)-(k+2)*b(k) is 1, 1, 0, -6, -30, -90, 0, 2520, 22680, 113400, 0, -7484400, -97297200, a version of A009775. This is also related to A090932. ]


From (16)

a(n)=2^n*(n+1)!*sum(k=0,n,[1,1,0,-1,-1,0][k%6+1]/(k+1)/2^k)

1, 5, 30, 234, 2316, 27792, 389808, 6241968, 112355424, 2246745600, 49424774400, 1186194585600,

with a(n)/(2^n*(n+1)!) -> 2*Pi*sqrt(3)/9


From (13)

b(n)=(n+1)!*sum(k=0,n,sin((k+1)*Pi/4)/(k+1)/2^((k-1)/2))

1, 3, 10, 40, 194, 1134, 7848, 62784, 567576, 5698440, 62796240, 753554880,

with b(n)/(n+1)! -> Pi/2


Combining both,

9*a(n)/(4*b(n)*2^n) -> sqrt(3)


Changing sin by cos in b(n),

b(n)=(n+1)!*sum(k=0,n,cos((k+1)*Pi/4)/(k+1)/2^((k-1)/2))

1, 2, 5, 17, 79, 474, 3408, 27894, 253566, 2535660, 27778860, 332098920,


Two more variants:

b(n)=(n+1)!*sum(k=0,n,sin(k*Pi/4)/(k+1)/2^((k-2)/2))

0, 1, 5, 23, 115, 660, 4440, 34890, 314010, 3162780, 35017380, 421455960,


b(n)=(n+1)!*sum(k=0,n,cos(k*Pi/4)/(k+1)/2^((k-2)/2))

2, 5, 15, 57, 273, 1608, 11256, 90678, 821142, 8234100, 90575100, 1085653800,