User:Jaume Oliver Lafont

Engineer. Started contributing to the OEIS in 2007, in sequence A058962.

BBP formulas

• A154920 Denominators of a ternary BBP-type formula for log(3)
• A165998 Denominators of Taylor series expansion of 1/(3*x)*log((1+x)/(1-x)^2)
• A164985 Denominators of ternary BBP-type series for log(5)
• A166486 Periodic sequence [0,1,1,1] of length 4
• A165132 Primes whose logarithms are known to possess ternary BBP formulas

• P(1,b,2,(1,0))

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

Other binary BBP formulas

Permutations of integers

• A166711 Sum_{k>0} 1/a(k) = log(2)
• A166871 Sum_{k>0} 1/a(k) = log(3/2)

General expressions for log(p/q) appear in the sequences.

Table of logarithms

As generalized Mercator series (or BBP-type formulas in base 1):

$log(1)=\left({\frac {0}{1}}\right)+\left({\frac {0}{2}}\right)+...$

$log(2)=\left({\frac {1}{1}}-{\frac {1}{2}}\right)+\left({\frac {1}{3}}-{\frac {1}{4}}\right)+...$

$log(3)=\left({\frac {1}{1}}+{\frac {1}{2}}-{\frac {2}{3}}\right)+\left({\frac {1}{4}}+{\frac {1}{5}}-{\frac {2}{6}}\right)+...$

$log(4)=\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}-{\frac {3}{4}}\right)+\left({\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}-{\frac {3}{8}}\right)+...$

$log(5)=\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {4}{5}}\right)+\left({\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}-{\frac {4}{10}}\right)+...$

$\dots$

Equivalently, as permutations of the harmonic series minus itself:

$log(1)=\left({\frac {1}{1}}-{\frac {1}{1}}\right)+\left({\frac {1}{2}}-{\frac {1}{2}}\right)+...$

$log(2)=\left({\frac {1}{1}}+{\frac {1}{2}}-{\frac {1}{1}}\right)+\left({\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{2}}\right)+...$

$log(3)=\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{1}}\right)+\left({\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}-{\frac {1}{2}}\right)+...$

$log(4)=\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{1}}\right)+\left({\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}-{\frac {1}{2}}\right)+...$

$log(5)=\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{1}}\right)+\left({\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}+{\frac {1}{10}}-{\frac {1}{2}}\right)+...$

$\dots$

Riemann series theorem makes this possible.

Series for $e^{\pi }$

• Sequence A166748 and comment to A039661.

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

Approximations to $e$ and $\pi$

$e\approx \left(\sum _{k=1}^{8}{\frac {1}{k}}\right)\left(1+{\frac {1}{80^{2}}}\right)=2.7182818(080)$

$\pi \approx \left(\sum _{k=1}^{12}{\frac {1}{k}}\right)\left(1+{\frac {1}{9^{2}}}\right)=3.1415(219)$

$\pi \approx 4-\left({\dfrac {2}{5}}\right)^{\frac {1}{6}}=3.141(6)$

Rational recurrences

• The family including A165903 and A165896 was studied in Re:A family of quadratic recurrences by Andrew Hone.
• A165904 and A165905 are Somos-4 variants.

Lucas numbers and log(2)

• Comment to sequences A000032 and A002162.

References

Some of these results have been included in the following works.

• David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants (formulas 44, 46, 47, 48, 49, 54 and 55 -these numbers need to be updated-)
• Richard J. Mathar, arXiv:1207.5845 Yet Another Table of Integrals] (formulas 0.179, 0.180, 0.183, 0.184, 0.191, 0.192, 0.193, 0.194, 1.39 -these numbers need to be updated-)
• Eric W. Weisstein, e Approximations (formula 9)

Notebook

• Dalzell-type integrals (Apr 2017)
• Sum of inverses (May 2013)
• Topics
• Most sequences
• Factorials links A000142, A000166 and A003048
• Harmonic series
• Pi supertask
• Pi includes several new sequences with properties.
• Constants Integrals and series for $\log(2)$, $\pi$, $(\log(2))^{2}$, $\pi ^{2}$
• A000297
• A000796 Decimal expansion of Pi
• PARI
• Zero relations
• BBP binary and ternary formulas
• 2^n mod N
• Sum of Reciprocals of Tetrahedral Numbers
• C((d+1)n+d,d)
• gamma
• arxiv