A078333 -id:A078333 - 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

A109259 a(n) = floor(n*sqrt(2)^sqrt(2)).

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 13, 14, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78, 79, 81, 83, 84, 86, 88, 89, 91, 93, 94, 96, 97, 99, 101, 102, 104

Offset: 1

Reinhard Zumkeller, Jun 23 2005

nonn

Beatty sequence for sqrt(2)^sqrt(2) = 1.63252...; complement of A109260.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A078333, A109261, A038127.

[Floor(n*Sqrt(2)^Sqrt(2)): n in [1..100]]; // G. C. Greubel, Mar 27 2018

With[{c=(Sqrt[2])^Sqrt[2]},Floor[c*Range[100]]] (* Harvey P. Dale, Mar 19 2018 *)

for(n=1,100, print1(floor(n*sqrt(2)^sqrt(2)), ", ")) \\ G. C. Greubel, Mar 27 2018

A185094 Decimal expansion of sqrt(3)^sqrt(3).

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 04 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002193, A078333

Mathematica
```
RealDigits[N[Sqrt[3]^Sqrt[3],200]][[1]]
```

3^(sqrt(3)/2) \\ Charles R Greathouse IV, Oct 01 2012

A194348 Decimal expansion of sqrt(2)^sqrt(2)^sqrt(2).

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Aug 28 2011

If Schanuel's Conjecture is true, then sqrt(2)^sqrt(2)^sqrt(2) is transcendental (see Marques and Sondow 2010, p. 79).

			1.76083955588002809075664989563837274807980409431850990464638822505342...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 1.1
D. Marques and Jonathan Sondow, Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental, arXiv:1010.6216 [math.NT], 2010-2011; East-West J. Math., 12 (2010), 75-84.
Wikipedia, Schanuel's conjecture

Cf. A002193 (decimal expansion of sqrt(2)), A078333 (decimal expansion of sqrt(2)^sqrt(2)), A194555 (the decimal expansion of the real part of I^e^Pi).

SetDefaultRealField(RealField(100)); Sqrt(2)^Sqrt(2)^Sqrt(2); // G. C. Greubel, Aug 19 2018

RealDigits[ Sqrt[2]^Sqrt[2]^Sqrt[2], 10, 100] // First

sqrt(2)^sqrt(2)^sqrt(2) \\ Charles R Greathouse IV, May 14 2014

(x->x^x^x)(sqrt(2)) \\ Charles R Greathouse IV, May 14 2014

A109261 Self-inverse integer permutation induced by Beatty sequences for sqrt(2)^sqrt(2) and sqrt(2)^sqrt(2)/(sqrt(2)^sqrt(2)-1).

Original entry on oeis.org

2, 1, 5, 7, 3, 10, 4, 12, 15, 6, 18, 8, 20, 23, 9, 25, 28, 11, 30, 13, 33, 36, 14, 38, 16, 41, 43, 17, 46, 19, 49, 51, 21, 54, 56, 22, 59, 24, 61, 64, 26, 67, 27, 69, 72, 29, 74, 77, 31, 80, 32, 82, 85, 34, 87, 35, 90, 92, 37, 95, 39, 98, 100, 40, 103

Offset: 1

Reinhard Zumkeller, Jun 23 2005

nonn

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to Beatty sequences

Cf. A109259, A109260.

Cf. A078333 (sqrt(2)^sqrt(2)).

a(A109259(n))=A109260(n) and a(A109260(n))=A109259(n).

A185110 Decimal expansion of sqrt(3)^sqrt(2).

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 04 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002193, A078333

Mathematica
```
RealDigits[N[Sqrt[3]^Sqrt[2],200]][[1]]
```

3^(sqrt(2)/2) \\ Charles R Greathouse IV, Oct 01 2012

A185111 Decimal expansion of sqrt(2)^sqrt(3).

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 04 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002193, A078333, A185094, A185110

Mathematica
```
RealDigits[N[Sqrt[2]^Sqrt[3],200]][[1]]
```

2^(sqrt(3)/2) \\ Charles R Greathouse IV, Oct 01 2012

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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A109259 a(n) = floor(n*sqrt(2)^sqrt(2)).

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A185094 Decimal expansion of sqrt(3)^sqrt(3).

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A194348 Decimal expansion of sqrt(2)^sqrt(2)^sqrt(2).

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A109261 Self-inverse integer permutation induced by Beatty sequences for sqrt(2)^sqrt(2) and sqrt(2)^sqrt(2)/(sqrt(2)^sqrt(2)-1).

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Formula

A185110 Decimal expansion of sqrt(3)^sqrt(2).

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PARI

A185111 Decimal expansion of sqrt(2)^sqrt(3).

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PARI