A156590 -id:A156590 - cp's OEIS frontend

A156548 Decimal expansion of the real part of the limit of f(f(...f(0)...)) where f(z)=sqrt(i+z).

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

Offset: 1

Clark Kimberling, Feb 12 2009

The imaginary part, 0.624810..., is given by A156590.

(a-1) is the limit of the real part of the same expression, but with f(z)=i/(1+z), and therefore the real part of the continued fraction i/(1+i/(1+i/(...))). Moreover, (a-1) equals also the imaginary part of the continued fraction i/(i+i/(i+i/(...))). - Stanislav Sykora, May 27 2015

			1.30024259022012041915890982074952138854853281918394761...

Cf. A156590.

RealDigits[1/2 + Sqrt[(1+Sqrt[17])/8],10,120][[1]] (* Vaclav Kotesovec, May 28 2015 *)

Define z(1)=f(0)=sqrt(i), where i=sqrt(-1), and z(n)=f(z(n-1)) for n>1.
Write the limit of z(n) as a+bi where a and b are real. Then a=(b+1)/(2b), where b=sqrt((sqrt(17)-1)/8).
Equals real part of 1/2 + Sum_{n>=0} ((-1)^(n/2 + 5/4)*binomial(2*n, n))/(2^(4*n)*(2*n - 1)). - Antonio Graciá Llorente, Nov 20 2024

A272876 Decimal expansion of the imaginary part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 15 2016

The real part and the modulus of this complex constant are in A272875 and A272877, respectively. For more information, see A272875.

			0.40756393054562184473966324339415208864062799286677510304874835...

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Cf. A156590, A272875, A272877.

RealDigits[Im[z /. FindRoot[(1 + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 26 2023 *)

\\ f(x) computes (x+(x+...)^i)^i, provided that it converges:
f(x)={my(z=1.0,zlast=0.0,eps=10.0^(1-default(realprecision)));while(abs(z-zlast)>eps,zlast=z;z=(x+z)^I);return(z)}
\\ To compute this constant, use:
z0 = f(1); imag(z0)

A347178 Decimal expansion of imaginary part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 21 2021

			0.3411639019140096636847418698555241284...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A060006, A156548, A156590, A347177.

RealDigits[((29 + 3 Sqrt[93])^(1/3) - 2^(1/3))/(2 (3 (9 + Sqrt[93]))^(1/3)), 10, 110][[1]]

((29 + 3*sqrt(93))^(1/3) - 2^(1/3))/(2*(3*(9 + sqrt(93)))^(1/3)) \\ Michel Marcus, Aug 21 2021

Equals -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3)/sqrt(3). - Vaclav Kotesovec, Sep 29 2024

A347177 Decimal expansion of real part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

This is the sum of the imaginary parts of the complex roots of the cubic equation 8*r^3 + 2*r - 1 = 0 , and its real solution is A347178. - Gerry Martens, Apr 02 2024

			1.1615413999972519360879176872471740748431...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A060006, A156548, A156590, A347178.

RealDigits[(1/12) 2^(2/3) 3^(5/6) ((Sqrt[93] - 9)^(1/3) + (9 + Sqrt[93])^(1/3)), 10, 110][[1]]

(1/12)*2^(2/3)*3^(5/6)*((sqrt(93) - 9)^(1/3) + (9 + sqrt(93))^(1/3)) \\ Michel Marcus, Aug 21 2021

2*imag(polroots(8*x^3 + 2*x - 1)[3]) \\ Gerry Martens, Apr 02 2024

Equals cosh(asinh(3*sqrt(3)/2)/3). - Gerry Martens, Apr 02 2024

A257945 Decimal expansion of abs(i/(i + i/(i + i/...))) and abs(i/(1 + i/(1 + i/...))), i being the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 29 2015

Set v = A156590 and u = (A156548 - 1). Then the continued fractions evaluate to i/(i + i/(i + i/...)) = (sqrt(4*i - 1) - i)/2 = v + u*i and i/(1 + i/(1 + i/...)) = (sqrt(4*i + 1) - 1)/2 = u + v*i. They can be evaluated either explicitly or as limits of the convergent recursive mappings z -> i/(i + z) and z -> i/(1 + z), respectively, starting, for example, with z = 0.

An algebraic integer of degree 8. - Charles R Greathouse IV, Jun 02 2015

			0.69320546462379732043436370422413868794102175016921901339955586752...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Index entries for algebraic numbers, degree 8

Cf. A156548, A156590.

RealDigits[Sqrt[1 + Sqrt[17] - Sqrt[2*(1 + Sqrt[17])]]/2, 10, 105][[1]] (* Vaclav Kotesovec, Jun 02 2015 *)

sqrt(1+sqrt(17)-sqrt(2*(1+sqrt(17))))/2

polrootsreal(x^8-x^6-2*x^4-x^2+1)[3] \\ Charles R Greathouse IV, Jun 02 2015

Equals sqrt(1 + sqrt(17) - sqrt(2*(1 + sqrt(17))))/2.

cp's OEIS Frontend

A156548 Decimal expansion of the real part of the limit of f(f(...f(0)...)) where f(z)=sqrt(i+z).

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A272876 Decimal expansion of the imaginary part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.

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A347178 Decimal expansion of imaginary part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

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A347177 Decimal expansion of real part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

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A257945 Decimal expansion of abs(i/(i + i/(i + i/...))) and abs(i/(1 + i/(1 + i/...))), i being the imaginary unit.

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