A277435 -id:A277435 - 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

A260691 Decimal expansion of a constant related to asymptotic behavior of super-roots of 2: lim_{n->inf} (sr[n](2) - sqrt(2))/log(2)^n.

Original entry on oeis.org

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

Offset: 0

Vladimir Reshetnikov, Nov 15 2015

Tetration is defined recursively: x^^0 = 1, x^^n = x^(x^^(n-1)). Its inverse, super-root, is defined: sr[n](y) = x iff x^^n = y. Note that lim_{n->inf} sr[n](2) = sqrt(2). Asymptotically, sr[n](2) = sqrt(2) + O(log(2)^n). This constant is the coefficient in the O(log(2)^n) term, i.e. lim_{n->inf} (sr[n](2) - sqrt(2))/log(2)^n.

			0.0685756598113291039765533114155...

G. C. Greubel, Table of n, a(n) for n = 0..193
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Super-root

Cf. A198094, A277435.

{0}~Join~RealDigits[SequenceLimit[1`200 Table[(2 - Power @@ Table[Sqrt[2], {n}])/Log[2]^n, {n, 1, 200}]] (1 - Log[2])/(2 Sqrt[2]), 10, 100][[1]] (* Vladimir Reshetnikov, Oct 18 2016 *)

a = A277435*(1-log(2))/(2*sqrt(2)). - Vladimir Reshetnikov, Oct 18 2016

A280918 2nd term of the continued fraction for 2-sqrt(2)^^n, where x^^n denotes tetration.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 20, 29, 42, 61, 88, 128, 184, 267, 385, 556, 803, 1159, 1672, 2413, 3481, 5023, 7247, 10456, 15085, 21764, 31399, 45299, 65354, 94286, 136026, 196245, 283122, 408459, 589282, 850155, 1226515, 1769487, 2552830, 3682956, 5313383, 7665592

Offset: 1

Vladimir Reshetnikov, Jan 10 2017

nonn

Tetration x^^n is defined recursively: x^^0 = 1, x^^n = x^(x^^(n-1)). Note that lim_{n->inf} sqrt(2)^^n = 2. This sequence shows the speed of convergence to this limit.

G. C. Greubel, Table of n, a(n) for n = 1..300
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration

Cf. A198094, A277435.

Table[ContinuedFraction[2 - Power@@Table[Sqrt[2], {n}], 2][[2]], {n, 42}]

a(n) ~ 1/(A277435*log(2)^n).

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.

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

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A260691 Decimal expansion of a constant related to asymptotic behavior of super-roots of 2: lim_{n->inf} (sr[n](2) - sqrt(2))/log(2)^n.

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A280918 2nd term of the continued fraction for 2-sqrt(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)^(2*n)).

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A288607 Exponential reversion of A288606.

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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)).