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

Author

Vladimir Reshetnikov, Oct 30 2011

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A002193, A078333, A194348.

Cf. A260691, A277435.

Programs

Mathematica

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

PARI

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

Formula

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