A194348 -id:A194348 - 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

A194555 Decimal expansion of the real part of i^(e^Pi), where i = sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Aug 28 2011

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

			i^e^Pi = 0.2192048949... - 0.9756788478...*i

G. C. Greubel, Table of n, a(n) for n = 0..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. A039661 (e^Pi), A194554 (imaginary part).

Cf. A194348 (sqrt(2)^(sqrt(2)^sqrt(2))).

RealDigits[ Re[I^E^Pi], 10, 100] // First

real(I^(exp(Pi))) \\ Michel Marcus, Aug 19 2018

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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