%I A046859
%S A046859 1,3,7,61
%N A046859 Simplified Ackermann function (main diagonal)
%C A046859 Another version of the Ackermann numbers is the sequence 1^1, 2^^2, 3^^^3, 
               4^^^^4, 5^^^^^5, ..., which begins 1, 4, 3^3^3^... (where the number 
               of 3's in the tower is 3^3^3 = 7625597484987), ... [Conway and Guy]. 
               This grows too rapidly to have its own entry in the OEIS.
%C A046859 An even more rapidly growing sequence is the Conway-Guy sequence 1, 2->
               2, 3->3->3, 4->4->4->4, ..., which agrees with the sequence in the 
               previous comment for n <= 3, but then the 4-th term is very much 
               larger than 4^^^^4.
%C A046859 The original Ackermann function f is defined by f(0,x,y)=y+1, f(1,x,y)=x+y, 
               f(2,x,y)=x*y, f(3,x,y)=x^y, etc.
%D A046859 W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 
               99 (1928), 118-133.
%D A046859 Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 
               p. 60, 1996.
%D A046859 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A046859 H. Hermes, Aufzaehlbarkeit, Entscheidbarkeit, Berechenbarkeit: Einfuehrung 
               in die Theorie der rekursiven Funktionen (3rd ed., Springer, 1978), 
               83-89.
%D A046859 H. Hermes, ditto, 2nd ed. also available in English (Springer, 1969), 
               ch. 13
%H A046859 W. Ackermann, <a href="http://134.76.163.65/servlet/digbib?template=view.html&id=29309&startpage=122&endpage=\
               137&image-path=http://134.76.176.141/cgi-bin/letgifsfly.cgi&image-subpath=/
               1417&image-subpath=1417&pagenumber=122&imageset-id=1417">Zum Hilbertschen 
               Aufbau der reellen Zahlen</a>, Math. Ann. 99 (1928), 118-133.
%F A046859 A(0, y) := y+1, A(x+1, 0) := A(x, 1), A(x+1, y+1) := A(x, A(x+1, y))
%Y A046859 Cf. A001695, A014221.
%Y A046859 Sequence in context: A100772 A131652 A164895 this_sequence A084289 A077703 
               A134705
%Y A046859 Adjacent sequences: A046856 A046857 A046858 this_sequence A046860 A046861 
               A046862
%K A046859 nonn,bref
%O A046859 0,2
%A A046859 D. E. Knuth
%E A046859 Next term is 2^(2^(2^(2^16))) - 3.
%E A046859 Additional comments from Frank.Ellermann(AT)t-online.de, Apr 21, 2001