来自在线整数百科全书的问候语!1~(-1)1αi %A0468 59 % S S A0468 59 1,3,7,61%N N A0468 59简化阿克曼函数(Akman)的主要对角线。%C A0468 59下一项为2 ^(2 ^(2 ^(2 ^ 16)))-3,这太大,不能在数据线中显示。http://oei.org/y*搜索:ID:A0468 59显示(塔中3的数量是3 ^ 3 ^ 3=7625597484987),…[康威和盖伊]。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 4th term is very much larger than 4^^^^4. %C A046859 From _Natan Arie' Consigli_, Apr 10 2016: (Start) %C A046859 A189896(n) = succ(0), 1+1, 2*2, 3^3,..., also called Ackermann numbers, is a weaker version of the above sequence. %C A046859 The Ackermann functions are well-known to be simple examples of computable (implementable using a combination of while/for-loops) but not primitive recursive (implementable using only a FINITE number of do-while/for-loops) functions. %C A046859 See A054871 for the definitions of the hyperoperations (a[n]b and H_n(a,b)). %C A046859 The original Ackermann function f is defined by: %C A046859 { %C A046859 {f(0,y,z)=y+z; %C A046859 {f(1,y,0)=0; %C A046859 {f(2,y,0)=1; %C A046859 {f(x,y,0)=x; %C A046859 {f(x,y,z)=f(x-1,y,f(x,y,z-1)) %C A046859 { %C A046859 Here we have f(1,y,z)=y*z, f(2,y,z)=y^z. %C A046859 Ackermann function variants are 3-argument functions that satisfy the recurrence relation above. %C A046859 Example: %C A046859 the hyperoperation function H(x,y,z) satisfies the original's recurrence relation but has the following initial values: %C A046859 { %C A046859 {H(0,y,z) = y+1; %C A046859 {H(1,y,0) = y; %C A046859 {H(2,y,0) = 0; %C A046859 {H(n,y,0) = 1. %C A046859 { %C A046859 The family of Ackermann functions can be simplified by omitting the "y" variable of the 3-argument function by making them have two arguments. %C A046859 A 2-argument Ackermann function would then be a function satisfying the recurrence relation: f(x,z)=f(x-1,f(x,z-1)). %C A046859 The most popular example is Ackermann-Péter's function defined by: %C A046859 { %C A046859 {A(0,y) = y+1; %C A046859 {A(x+1,0) = A(x,1); %C A046859 {A(x+1,y+1) = A(x,A(x+1,y)) %C A046859 { %C A046859 Here we have A(0,y-1) = y = 2[0](y-1+3)-3. %C A046859 Suppose A(x-1,y-1) = 2[x-1](y-1+3)-3. %C A046859 By induction on positive x: %C A046859 since 2[x]2 = 4 (See A255176) we have A(x,0) = A(x-1,1) = 2[x-1]4-3 = 2[x-1]2[x-1]2-3 = 2[x-1]3-3. %C A046859 By induction on positive y we can conclude that: %C A046859 A(x,y) = A(x-1,A(x,y-1)) = 2[x-1](2[x](y-1+3)-3+3)-3 = 2[x-1]2[x](y-1+3)-3 = 2[x](y+3)-3. %C A046859 * %C A046859 If f is a 3-argument (2-argument) Ackermann function, Ack(n) = f(n,n,n) (f(n,n)) is called a simplified Ackermann function. “阿克曼数”是ACK(n).0%C A0468 59的值,这里我们有一个(n)=a(n,n)=2 [n](n+1)- 3,η%ca0468 59(结尾)%d a0468 59康威,J. H.和盖伊,R. K.,数字书。纽约:Springer Verlag,第60, 1996页.0%D A0468 59.珠穆朗玛峰,A.Van Del-Poeltn,I. Shparlinski和T. Ward,递推序列,阿梅尔。数学。SoC,2003;参见第255页。爱马仕,Aufzaehlbarkeit,Entscheidbarkeit,BeleChanbBaKIT:EnFueHung在EnfusieRunn Funksivin FunkTunn(第三版,Springer,1978),83-89.0%D A0468 59。h,爱马仕,DITTO,第二版。英文版(SpRIGER,1969),CH 13 %%H A0468 59 W. Ackermann,我爱你数学。安。99(1928),118-133.0%H A0468 59 D. E. Knuth和N.J.A.斯隆,1970年5月通讯%F A0468 59从S.Nahan-AARE’CuxigLi],4月10日2016:(开始)F 0468 59 A(0,Y):=Y+ 1,A(x+1, 0):=a(x,1),a(x+1,y+1):=a(x,a(x+1,y));ηf a0468 59 a(n)=a(n,n).f %a0468 59 A(n)=2 [n](n+3)-3=Hyn(2,n+3)-^。%E A0468 59 A(0)=2〔0〕(0+3)-3=1;εE A0468 59 A(1)=2〔1〕(1+1)=εE A0468 59 A(α)=ω[α](α+)-α=γ;% %E A0468 59 A(α)=α[α](α+)-α=γ;% %E A0468 59 A(α)=α[α](α+)-α=^((^ ^))-^。(2016)4月10日,(NeXT)E.A0468 59:(结束)%AY A0468 59参见A0599 36,A266200,A27 1553。(sequences involving simplified Ackermann Functions) %Y A046859 Cf. A001695, A014221, A143797, A264929 (sequences involving other versions of two-argument Ackermann's Function). %Y A046859 Cf. A054871, A189896 (sequences involving variants of the three-argument Ackermann's Function). %Y A046859 Cf. A126333 (a(n)=A(n,0)), A074877 (a(n)=A(3,n)). %Y A046859 Cf. A260002-A260006 (sequences with Sudan's function, another computable but not primitive recursive function). %Y A046859 Cf. A266201 (Goodstein's function, total and not primitive recursive). %K A046859 nonn,bref %O A046859 0,2 %A A046859 _Don Knuth_ %E A046859 Additional comments from _Frank Ellermann_, Apr 21 2001 %E A046859 Name clarified by _Natan Arie' Consigli_, May 13 2016 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE