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 A244109 Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers. 5
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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*sumalt(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,changed AUTHOR Jean-François Alcover, Jun 20 2014 EXTENSIONS Corrected by Vaclav Kotesovec, Dec 11 2015 STATUS approved

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Last modified June 18 01:14 EDT 2021. Contains 345092 sequences. (Running on oeis4.)