A113024 -id:A113024 - cp's OEIS frontend

A337998 Decimal expansion of Sum_{n>=1} (cos((nPi)/2) + sin((nPi)/2)) / sqrt(n).

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

Offset: 0

Peter Luschny, Nov 03 2020

			0.2399635244956309553375743101605772258978644368017700426676289374...

zeta(1/2, 1/k): A059750 (k=1), A113024 (k=2), this sequence (k=4).

evalf(Zeta(0, 1/2, 1/4)*10^86, 100):
ListTools:-Reverse(convert(floor(%), base, 10));

RealDigits[Zeta[1/2, 1/4], 10, 100][[1]] (* Vaclav Kotesovec, Nov 03 2020 *)

Equals zeta(1/2, 1/4).

A362742 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*floor(sqrt(k))/k.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 02 2023

If the floor function is replaced by the fractional part function, then Sum_{k>=1} (-1)^(k+1)*frac(sqrt(k))/k = (A113024 - (this constant)) = 0.01333786407...

			0.591560779349817340213846903345...

Wolfgang Hintze and River Li, Closed expression for sum Sum_{k=1..oo} (-1)^(k+1)*floor(sqrt(k))/k, Mathematics Stackexchange, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.

Cf. A000196, A113024.

evalf(log(2) + Sum((-1)^n*n*Sum(1/((n^2 + 2*i - 1)*(n^2 + 2*i)), i = 1..n), n = 1..infinity), 200); # Vaclav Kotesovec, May 02 2023

RealDigits[NIntegrate[(1 - EllipticTheta[4, x])/(2*x*(x + 1)), {x, 0, 1}, WorkingPrecision -> 30]][[1]]

default(realprecision, 200); log(2) + sumalt(n=1, (-1)^n*n*sum(i=1, n, 1/((n^2 + 2*i - 1)*(n^2 + 2*i)) )) \\ Vaclav Kotesovec, May 02 2023

Equals log(2) + Sum_{n>=1} (-1)^n*n*Sum_{i=1..n} 1/((n^2+2*i-1)*(n^2+2*i)) (Li, 2019).

Equals Integral_{x=0..1} (1-theta_4(0,x))/(2*x*(x+1)), where theta_4(z, q) is the 4th Jacobi theta function (Hintze, 2019).

More digits from Vaclav Kotesovec, May 02 2023

A367409 Decimal expansion of arclength of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 26 2023

See A367309.

			1.0186563516740513673662299252527545340266...

Cf. A113024, A367309, A367311.

f[x_] := (1 - 2^(1 - x)) Zeta[x]
y = NIntegrate[Sqrt[1 + f'[x]^2], {x, 0, 1}, WorkingPrecision -> 200]
RealDigits[y][[1]]

f(x) = (1 - 2^(1-x))*zeta(x); intnum(x=0, 1, sqrt(1+f'(x)^2)) \\ Michel Marcus, Nov 27 2023

A380547 Decimal expansion of the absolute value of the sum of the Dirichlet L-series A000035 at s=1/2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 26 2025

Defined as L(s) = (1-2^(-s))*zeta(s) by analytic continuation of the Riemann zeta function.

			Sum_{n>=1} A000035(n)/sqrt(n) = -0.42772793269397822132111661913967125635373339294116...

Cf. A111003 (s=2), A233091 (s=3), A300707 (s=4), A059750 (zeta(1/2)), A000035, A010503, A113024, A268682.

RealDigits[(1/Sqrt[2]-1)*Zeta[1/2], 10, 120][[1]] (* Amiram Eldar, Jan 26 2025 *)

(1/sqrt(2)-1)*zeta(1/2) \\ Amiram Eldar, Jan 26 2025

Equals A268682*A059750.

Equals A010503 * A113024 = Sum_{n>=1} (-1)^(n+1)/sqrt(2*n). - Amiram Eldar, Jan 26 2025

cp's OEIS Frontend

A337998 Decimal expansion of Sum_{n>=1} (cos((n*Pi)/2) + sin((n*Pi)/2)) / sqrt(n).

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A362742 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*floor(sqrt(k))/k.

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A367409 Decimal expansion of arclength of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

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A380547 Decimal expansion of the absolute value of the sum of the Dirichlet L-series A000035 at s=1/2.

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A337998 Decimal expansion of Sum_{n>=1} (cos((nPi)/2) + sin((nPi)/2)) / sqrt(n).