|
|
A157332
|
|
Denominators of Egyptian fraction for Pi/16 based on Machin's formula
|
|
4
|
|
|
5, -956, -375, 163823028, 15625, -15596225303980, -546875, 1247220779824098212, 17578125, -91597497639855832244124, -537109375, 6394838587727583881086964116, 15869140625, -431694043145875922302762745864588, -457763671875
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
Machin's formula: Pi/4 = 4*atan(1/5) - atan(1/239).
Sum_{n>=0} 1/a(n) = Pi/16 = atan(1/5) - (1/4)*atan(1/239).
|
|
LINKS
|
|
|
FORMULA
|
a(2n) = (2*n+1)*5^(2*n+1)*(-1)^n,
a(2n+1) = -4*(2*n+1)*239^(2*n+1)*(-1)^n.
G.f.: 5*(1-25*x^2)/(1+25*x^2)^2 - 956*x*(1-57121*x^2)/(1+57121*x^2)^2
|
|
MAPLE
|
seq(coeff(series(5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2, x, n+1), x, n), n = 0..15); # G. C. Greubel, Aug 26 2019
|
|
MATHEMATICA
|
CoefficientList[Series[5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2, {x, 0, 15}], x] (* G. C. Greubel, Aug 26 2019 *)
|
|
PROG
|
(PARI) my(x='x+O('x^15)); Vec(5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2) \\ G. C. Greubel, Aug 26 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 15); Coefficients(R!( 5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2 )); // G. C. Greubel, Aug 26 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 5*(1-(5*x)^2)/(1+(5*x)^2)^2 - 4*239*x*(1-(239*x)^2)/(1+(239*x)^2)^2 ).list()
(GAP)
a:= function(n)
if n mod 2=0 then return (-1)^(n/2)*(n+1)*5^(n+1);
else return -4*(-1)^((n-1)/2)*n*(239)^n;
fi;
end;
|
|
CROSSREFS
|
|
|
KEYWORD
|
frac,sign,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|