A260691 -id:A260691 - cp's OEIS frontend

A198094 3rd term of continued fraction for sqrt(2)^sqrt(2)^...^sqrt(2) with n sqrt(2)'s.

2, 1, 3, 5, 8, 12, 19, 28, 41, 60, 87, 127, 183, 266, 384, 555, 802, 1158, 1671, 2412, 3480, 5022, 7246, 10455, 15084, 21763, 31398, 45298, 65353, 94285, 136025, 196244, 283121, 408458, 589281, 850154, 1226514, 1769486, 2552829, 3682955, 5313382

Offset: 1

Vladimir Reshetnikov, Oct 30 2011

1st terms are 1,1,1,1,1,... and 2nd terms are 2,1,1,1,1,...

G. C. Greubel, Table of n, a(n) for n = 1..251

Cf. A002193, A078333, A194348.

Cf. A260691, A277435.

ContinuedFraction[#, 3][[3]] & /@ NestList[Sqrt[2]^# &, Sqrt[2], 40]

a(n) = {my(c = sqrt(2)); for (k=1, n-1, c = sqrt(2)^c); contfrac(c)[3];} \\ Michel Marcus, Oct 19 2016

a(n) ~ c / log(2)^n, where c = 1/A277435 = 1.582031511247872306827383... - Vladimir Reshetnikov, Oct 18 2016

A277435 Decimal expansion of lim_{n->inf} (2 - sqrt(2)^^n)/log(2)^n, where x^^n denotes tetration.

Original entry on oeis.org

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

Offset: 0

Vladimir Reshetnikov, Oct 14 2016

Tetration x^^n is defined recursively: x^^0 = 1, x^^n = x^(x^^(n-1)). Note that sqrt(2)^^inf = lim_{n->inf} sqrt(2)^^n = 2. Asymptotically, sqrt(2)^^n = 2 - O(log(2)^n). This constant is the coefficient in the O(log(2)^n) term. Furthermore, sqrt(2)^^n = 2 - a*log(2)^n + (a^2/(4*(1 - 1/log(2))))*log(2)^(2*n) + O(log(2)^(3*n)).

			0.63209866105082925035545064599078...

Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration

Cf. A198094, A260691.

RealDigits[SequenceLimit[1`200 Table[(2 - Power @@ Table[Sqrt[2], {n}])/Log[2]^n, {n, 1, 200}]], 10, 100][[1]]

a = 2*sqrt(2)*A260691/(1-log(2)).

A288606 E.g.f. expansion of f(x) around x = 1, where f(x) is the coefficient from the tetration asymptotic: x^^n = x^^inf - f(x)log(x^^inf)^n + O(log(x^^inf)^(2n)).

Original entry on oeis.org

0, 1, 2, 6, 26, 120, 474, -3500, -169744, -4739628, -122528220, -3244006128, -89971866744, -2643601630488, -82449886989120, -2730313541889120, -95853665484598656, -3561107748108889344, -139703010646898138688, -5774800668716738596896, -250987866830927324395200

Offset: 0

Vladimir Reshetnikov, Jun 11 2017

sign

The tetration x^^n is defined recursively: x^^0 = 1, x^^n = x^(x^^(n-1)). For x in [e^(-e), e^(1/e)] there is a limit x^^inf = limit_{n->inf} x^^n = e^(-W(-log x)), where W(z) is the Lambert W-function. The tetration approaches this limit exponentially: x^^n = x^^inf - f(x)*log(x^^inf)^n + O(log(x^^inf)^(2*n)), where the coefficient f(x) = lim_{n->inf} (x^^inf - x^^n)/log(x^^inf)^n depends on x. This sequence gives the e.g.f. expansion of f(x) around x = 1.

			f(x) = (1/1!)*(x-1) + (2/2!)*(x-1)^2 + (6/3!)*(x-1)^3 + (26/4!)*(x-1)^4 + (120/5!)*(x-1)^5 + ...

MathOverflow, Discussion of a related sequence.
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Lambert W function.
Wikipedia, Tetration.

Cf. A198094, A260691, A277435, A288607.

a[n_] := n! SeriesCoefficient[(Exp[-ProductLog[-Log[x]]] - Power @@ Table[x, {n}])/(-ProductLog[-Log[x]])^n, {x, 1, n}]; Table[a[n], {n, 0, 20}]

A288607 Exponential reversion of A288606.

Original entry on oeis.org

0, 1, -2, 6, -26, 180, -1734, 23464, -385160, 7561308, -166580820, 4109707800, -110972371608, 3276276647280, -104668338898200, 3609072471039840, -133458102348679680, 5274992059017870048, -221831508056339323584, 9893765935654872310656, -466361328442205843665920

Offset: 0

Vladimir Reshetnikov, Jun 11 2017

sign

It appears that the terms have alternating signs, and their absolute values grow monotonically and are log-convex.

MathOverflow, Discussion of a related sequence.
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A198094, A260691, A277435, A288606.

cp's OEIS Frontend

A198094 3rd term of continued fraction for sqrt(2)^sqrt(2)^...^sqrt(2) with n sqrt(2)'s.

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A277435 Decimal expansion of lim_{n->inf} (2 - sqrt(2)^^n)/log(2)^n, where x^^n denotes tetration.

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A288606 E.g.f. expansion of f(x) around x = 1, where f(x) is the coefficient from the tetration asymptotic: x^^n = x^^inf - f(x)log(x^^inf)^n + O(log(x^^inf)^(2n)).

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A288607 Exponential reversion of A288606.

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cp's OEIS Frontend

A198094 3rd term of continued fraction for sqrt(2)^sqrt(2)^...^sqrt(2) with n sqrt(2)'s.

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Author

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Mathematica

PARI

Formula

A277435 Decimal expansion of lim_{n->inf} (2 - sqrt(2)^^n)/log(2)^n, where x^^n denotes tetration.

Views

Author

Keywords

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Mathematica

Formula

A288606 E.g.f. expansion of f(x) around x = 1, where f(x) is the coefficient from the tetration asymptotic: x^^n = x^^inf - f(x)*log(x^^inf)^n + O(log(x^^inf)^(2*n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A288607 Exponential reversion of A288606.

Views

Author

Keywords

Comments

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Crossrefs

A288606 E.g.f. expansion of f(x) around x = 1, where f(x) is the coefficient from the tetration asymptotic: x^^n = x^^inf - f(x)log(x^^inf)^n + O(log(x^^inf)^(2n)).