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OFFSET
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0,2
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COMMENT
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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.
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.
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.
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REFERENCES
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W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), 118-133.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 60, 1996.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
H. Hermes, Aufzaehlbarkeit, Entscheidbarkeit, Berechenbarkeit: Einfuehrung in die Theorie der rekursiven Funktionen (3rd ed., Springer, 1978), 83-89.
H. Hermes, ditto, 2nd ed. also available in English (Springer, 1969), ch. 13
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LINKS
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W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), 118-133.
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FORMULA
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A(0, y) := y+1, A(x+1, 0) := A(x, 1), A(x+1, y+1) := A(x, A(x+1, y))
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CROSSREFS
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Cf. A001695, A014221.
Sequence in context: A100772 A131652 A164895 this_sequence A084289 A077703 A134705
Adjacent sequences: A046856 A046857 A046858 this_sequence A046860 A046861 A046862
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KEYWORD
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nonn,bref
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AUTHOR
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D. E. Knuth
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EXTENSIONS
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Next term is 2^(2^(2^(2^16))) - 3.
Additional comments from Frank.Ellermann(AT)t-online.de, Apr 21, 2001
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