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A052110
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cc=the decimal expansion of limit c^c^c^c... (with an even number of terms) where c is the constant defined in A037077.
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1
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4, 6, 1, 9, 2, 1, 4, 4, 0, 1, 6, 4, 4, 1, 1, 4, 4, 5, 4, 0, 8, 5, 8, 8, 6, 4, 2, 6, 1, 4, 1, 9, 4, 5, 7, 8, 6, 3, 5, 0, 2, 8, 2, 8, 0, 1, 3, 6, 4, 8, 8, 2, 2, 8, 4, 4, 3, 4, 1, 6, 2, 9, 2, 7, 3, 5, 8, 9, 1, 7, 2, 5, 0, 2, 1, 4, 1, 5, 0, 1, 9, 5, 2, 8, 7, 5, 1, 9, 9, 4, 2, 2, 2, 5, 8, 7, 8, 6, 0, 4, 7, 3, 5, 7, 5
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| The limit c^c^c^c... (with an odd number of terms) where c is the constant defined in A037077 also converges to this value.
The divergent alternating series, defined in A037077, c = Sum[(-1)^n*n^(1/n),{n,1,Infinity}] has two limit points, 0.1878... and -0.8121..., as well as a Levin's u-transform sum of -0.3121...=1/2(-0.8121...+0.1878...).
The MRB constant is the upper limit point of Limit[Sum[(-1)^n*n^(1/n),{n,1,x}],x->Infinity], and the lower limit point is given by c1=MRB constant-1. Let c = MRB constant and d=abs(c1). Then notice that the iterative processes of c^c^c^… or d^d^d^… produces constants cc and dd respectively, and cc converges about 10 times slower than dd. [Marvin Ray Burns, Jul 03 2011]
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REFERENCES
| S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.
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LINKS
| S. R. Finch, Iterated Exponential Constants
Eric Weisstein's World of Mathematics, Power Tower
Gus Wiseman, Tetration
Wikipedia, Tetration
OEIS Wiki, Tetration
OEIS Wiki, MRB constant
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EXAMPLE
| =0.4619214401644114454085886426141945786350282801364882284434162927358917250...
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MATHEMATICA
| PowerTower[x_, n_]:=Nest[Power[x, #]&, x, n-1]; m=NSum[(-1)^n*(n^(1/n)-1), {n, Infinity}, WorkingPrecision -> 100, Method -> "AlternatingSigns"]; RealDigits[N[PowerTower[m, 860], 100]]
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PROG
| (PARI)
default(realprecision, 66);
M=sumalt(x=1, (-1)^x*((x^(1/x))-1));
solve(x=.46, .462, x^(1/x)-M)
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CROSSREFS
| Cf. A037077, A000027, A000312, A002488, A073230 .
Sequence in context: A199371 A156789 A195423 * A197020 A202321 A195425
Adjacent sequences: A052107 A052108 A052109 * A052111 A052112 A052113
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KEYWORD
| cons,nonn
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AUTHOR
| Marvin Ray Burns (bmmmburns(AT)sbcglobal.net) Jan 20 2000, Mar 28 2008, Nov 08 2009, Mar 24 2010, June 27, 2011
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