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))

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

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):

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

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

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

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

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

$\dots$

Equivalently, as permutations of the harmonic series minus itself:

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

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

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

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

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

$\dots$

Riemann series theorem makes this possible.

Series for $e^{\pi }$

• Sequence A166748 and comment to A039661.

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

Approximations to $e$ and $\pi$

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

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

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

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