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%I #60 Mar 19 2019 11:28:10
%S 1,1,4,7625597484987
%N Ackermann's sequence: n^^n := n^n^n^...^n (with n n's).
%C Using Knuth's arrow notation, this is n^^^2 (n-penta-2) or n^^n (n-tetra-n). - Andrew Robbins, Apr 16 2009
%C Comment from Trevor Green: The fourth term in this sequence has about as many digits - 8.07 * 10^153 - as the *square* of the number of protons in the universe.
%C We could prepend a(0) = 1 (since 0^^0 = 1, that is, the "empty power tower" gives the "empty product"). - _Daniel Forgues_, May 17 2013
%C The last 60 decimal digits of a(4) are ...67586985427238232605843019607448189676936860456095261392896. - _Daniel Forgues_, Jun 25 2016
%C From _Daniel Forgues_, Jul 06 2016: (Start)
%C a(4) has (the following number having 154 decimal digits)
%C 80723047260282253793826303970853990300713679217387 \
%C 43031867082828418414481568309149198911814701229483 \
%C 451981557574771156496457238535299087481244990261351117 decimal digits.
%C a(4) = 4^4^4^4 = 4^
%C 13407807929942597099574024998205846127479365820592 \
%C 39337772356144372176403007354697680187429816690342 \
%C 7690031858186486050853753882811946569946433649006084096,
%C the exponent of 4 having 155 decimal digits. (End)
%C The fractional part of 4^4^4*log[10](4) starts .373100157363599870..., so the first few digits of a(4) are 23610226714597313.... - _Robert Israel_, Jul 06 2016
%H W. Ackermann, <a href="http://eretrandre.org/rb/files/Ackermann1928_126.pdf">Zum Hilbertschen Aufbau der reellen Zahlen</a>, Math. Ann. 99 (1928), 118-133, DOI:10.1007/BF01459088.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AckermannNumber.html">Ackermann Number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Knuth's_up-arrow_notation">Knuth's up-arrow notation</a>
%p b:= (n, i)-> `if`(i=0, 1, n^b(n, i-1)):
%p a:= n-> b(n, n):
%p seq(a(n), n=0..3); # _Alois P. Heinz_, Aug 22 2017
%t a[n_] := If[n == 0, 1, Nest[n^#&, n, n-1]];
%t Table[a[n], {n, 0, 3}] (* _Jean-François Alcover_, Mar 19 2019 *)
%Y Cf. A027747, A008868 (which mentions an older estimate for the same number), A266200.
%Y Main diagonal of A321312.
%K nonn
%O 0,3
%A Daniel Wild (wild(AT)edumath.u-strasbg.fr)
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