A244109 Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers.

2, 0, 4, 6, 2, 7, 7, 4, 5, 2, 8, 5, 5, 8, 7, 8, 5, 9, 1, 0, 7, 0, 1, 7, 6, 1, 5, 3, 9, 5, 0, 4, 3, 6, 1, 9, 4, 9, 8, 4, 2, 9, 0, 5, 5, 8, 7, 3, 2, 1, 6, 6, 5, 1, 8, 7, 3, 2, 6, 9, 7, 2, 3, 5, 8, 2, 4, 3, 3, 0, 6, 3, 8, 4, 5, 7, 0, 4, 6, 5, 5, 7, 8, 4, 5, 5, 0, 6, 3, 9, 4, 4, 8, 2, 4, 3, 4, 1, 7, 5, 0, 0, 2, 1, 4

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Formula

Equals Sum_{k>=1} log(k*(k+1))/(k*(k+1)).

Equals A085361 + A131688. - Vaclav Kotesovec, Dec 11 2015

Equals Sum_{n >=1} ((1 + (-1)^(n+1))*zeta(n + 1) - 1)/n. - G. C. Greubel, Nov 15 2018

Equals 2*Sum_{k>=2} log(k)/(k^2-1) = 2*A340440. - Gleb Koloskov, May 02 2021

Equals -2*Sum_{k>=1} zeta'(2*k). - Vaclav Kotesovec, Jun 17 2021

Example

2.04627745285587859107017615395043619498429055873216651873269723582433...

Maple

evalf(Sum(((1 + (-1)^(n+1))*Zeta(n+1) - 1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015

Mathematica

NSum[Log[k*(k+1)]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *)
digits = 120; RealDigits[NSum[((1-(-1)^n)*Zeta[n+1] -1)/n, {n, 1, Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *)

Prog

(PARI) default(realprecision, 1000); s = sumalt(n=1, ((1 + (-1)^(n+1))*zeta(n+1) - 1)/n); default(realprecision, 100); print(s) \\ Vaclav Kotesovec, Dec 11 2015
(PARI) 2*suminf(k=1, -zeta'(2*k)) \\ Vaclav Kotesovec, Jun 17 2021
(Magma) SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[((1-(-1)^n)*Evaluate(L, n+1)-1)/n: n in [1..1000]]); // G. C. Greubel, Nov 15 2018
(Sage) numerical_approx(sum(((1-(-1)^k)*zeta(k+1)-1)/k for k in [1..1000]), digits=120) # G. C. Greubel, Nov 15 2018

Crossrefs

Cf. A002210, A085361. Equals twice A340440.

Sequence in context: A233673 A319931 A192133 * A133144 A342384 A192134

Adjacent sequences: A244106 A244107 A244108 * A244110 A244111 A244112

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 20 2014

Extensions

Corrected by Vaclav Kotesovec, Dec 11 2015

Status

approved