A156548 -id:A156548 - cp's OEIS frontend

A272875 Decimal expansion of the real part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.

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

Offset: 0

Stanislav Sykora, May 15 2016

The mapping M(z)=(1+z)^i has in C a unique invariant point, namely z0 = a+A272876*i, which is also its attractor. Iterative applications of M applied to any starting complex point z (except for the singular value -1+0*i) rapidly converge to z0. The convergence, and the existence of this limit, justify the expression used in the name. It is easy to show that, close to z0, the convergence is exponential, with the error decreasing approximately by a factor of abs(z0/(1+z0))=0.4571... per iteration.

The imaginary part and the modulus of this complex constant are in A272876 and A272877, respectively.

			0.6738813311078755157802311904681019338764503347933725454899813516...

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

Cf. A156548, A272876, A272877.

RealDigits[Re[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); real(z0)

z0 = a+A272876*i satisfies the equations (1+z0)^i = z0, (1+z0)*z0^i = 1.

A272877 Decimal expansion of the modulus of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 15 2016

The real and imaginary parts giving rise to this constant are in A272875 and A272876, respectively. For more information, see A272875.

			0.78754327239683701096766024053943642458945927728138840827609389960...

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

Cf. A156548, A272875, A272876.

\\ 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); abs(z0)

A156590 Decimal expansion of the imaginary part of the limit of f(f(...f(0)...)) where f(z)=sqrt(i+z).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Feb 12 2009

The real part, 1.300242590..., is given by A156548.

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

			0.6248105338...

Cf. A156548.

RealDigits[Sqrt[(Sqrt[17]-1)/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).

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

A272875 Decimal expansion of the real part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.

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A272877 Decimal expansion of the modulus of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.

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A156590 Decimal expansion of the imaginary part of the limit of f(f(...f(0)...)) where f(z)=sqrt(i+z).

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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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