User:Jaume Oliver Lafont/Zero relations

From log(2)

$0=\sum _{k=0}^{\infty }{\frac {1}{(4k+1)}}-{\frac {3}{(4k+2)}}+{\frac {1}{(4k+3)}}+{\frac {1}{(4k+4)}}$

(A176563)

From Pi/(3*sqrt(3))

$0=\sum _{k=0}^{\infty }{\frac {1}{(6k+1)}}-{\frac {3}{(6k+2)}}+{\frac {3}{(6k+4)}}-{\frac {1}{(6k+5)}}$

From log(2), log(3) and log(6)

$0=\sum _{k=0}^{\infty }{\frac {1}{(6k+1)}}-{\frac {1}{(6k+2)}}-{\frac {2}{(6k+3)}}-{\frac {1}{(6k+4)}}+{\frac {1}{(6k+5)}}+{\frac {2}{(6k+6)}}$

From Pi/4

$0=\sum _{k=0}^{\infty }{\frac {1}{(8k+1)}}-{\frac {2}{(8k+2)}}-{\frac {1}{(8k+3)}}+{\frac {1}{(8k+5)}}+{\frac {2}{(8k+6)}}-{\frac {1}{(8k+7)}}$

From Pi^2/6 and Pi^2/8

$0=\sum _{k=0}^{\infty }{\frac {1}{(2k+1)^{2}}}-{\frac {3}{(2k+2)^{2}}}$

(A177927)