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 A266200 Herbert-Ackermann numbers: a(n) = Hack(n,n,n) = H_{n+1}(n,n), where Hack is the Herbert-Ackermann function. 2
 0, 0, 1, 4, 7625597484987 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,4 COMMENTS The Herbert-Ackermann function is defined as follows:   Hack(0,y,z) := y+z;   Hack(x,y,0) := 0, x > 0;   Hack(x,y,1) := y, x > 0;   Hack(x,y,z) := Hack(x-1,y,Hack(x,y,z-1)), x > 0. This is an Ackermann function variant: the recurrence relation is also satisfied by the original (3-argument) Ackermann function. Write a[n]b as H_n(a,b), the n-th hyperoperator of a times b (See A054871 for more information). We have Hack(0,0,0) = 0 and, for x > 0, Hack(x,y,1) = y = y[x+1]1. Suppose Hack(x-1,y,z-1) = y[x](z-1). By induction on z, Hack(x,y,z) = Hack(x-1,y,Hack(x,y,z-1)) = y[x]y[x+1](z-1) = y[x+1]z; so Hack(n,n,n) = n[n+1]n for nonnegative n. LINKS Robert S. Boyer,Versions of Ackermann functions FORMULA a(-1) = 0; a(n) = Hack(n,n,n), n >= 0. a(n) = H_{n+1}(n,n) = n[n+1]n, n >= -1. EXAMPLE a(-1) = (-1)(-1) = 0; (successor of -1) a(0) = 00 = 0+0 = 0; a(1) = 11 = 1*1 = 1; a(2) = 22 = 2^2 = 4; a(3) = 33 = 3^^3 = 3^3^3 = 3^27 = 7625597484987; a(4) = 44 = 4^^^4 = 4^^4^^4^^4 = 4^^4^^(4^4^4^4) = ... (where 4^4^4^4 = 10^(8.0723... × 10^153), thus 4^^4^^(4^4^4^4) is humongous!) Recursively: a(0) = Hack(0,0,0) = 0+0 = 0; a(1) = Hack(1,1,1) = Hack(0,1,Hack(1,1,0)) = Hack(0,1,0) = 1+0 = 1; a(2) = Hack(2,2,2) = Hack(1,2,Hack(2,2,1)) = Hack(1,2,2) =   Hack(0,2,Hack(1,2,1)) = Hack(0,2,2) = 2+2 = 4; a(3) = Hack(3,3,3) = Hack(2,3,Hack(3,3,2)) = Hack(2,3,Hack(2,3,Hack(3,3,1))) = Hack(2,3,Hack(2,3,3)) = ... (the number of recursions is already exploding...) CROSSREFS Cf. Sequences involving 2-argument Ackermann function variants: A001695, A046859, A074877, A126333, A143796, A143797. For sequences involving 3-argument Ackermann function variants see A054871. Cf. A004231. Sequence in context: A164796 A324441 A004231 * A066546 A132653 A115544 Adjacent sequences:  A266197 A266198 A266199 * A266201 A266202 A266203 KEYWORD nonn AUTHOR Natan Arie' Consigli, Dec 23 2015 STATUS approved

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Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)