A264930 - OEIS

%I #17 Jan 11 2020 15:57:47

%S 1,2,4,27,340282366920938463463374607431768211456

%N a(n) = n<n>n where x<n>y is a right associative hyperoperation.

%C Define x<n>y recursively as follows (this is a right associative version):

%C x<0>y = x+1;

%C x<1>0 = x;

%C x<2>0 = 0;

%C x<n>y = ( x<n>(y-1) ) <n-1> x for n>0 and y>0.

%C We have:

%C x<1>y = (...((x<0>y)<0>y)...y)(y nested parenthesis) = x+y;

%C x<2>y = (...((x<1>y)<1>y)...y)(y nested parenthesis) = x*y;

%C x<3>y = (...((x<2>y)<2>y)...y)(y nested parenthesis)  = x^y;

%C x<4>y = (...((x<3>y)<3>y)...y)(y nested parenthesis) = (...((x^x)^x)...^x) (a height-y bottom-up tower power, NOT tetration) = x^x^(y-1).

%C ...

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperoperation">Hyperoperations</a>.

%e a(0) = 0+1 = 1;

%e a(1) = 1+1 = 2;

%e a(2) = 2*2 = 4;

%e a(3) = 3^3 = 27;

%e a(4) = ((4^4)^4)^4 = 340282366920938463463374607431768211456;

%e a(5) = 5<5>5 = (((((5<4>5)<4>5)<4>5)<4>5)<4>5)<4>5.

%Y Cf. A054871, A189896.

%K nonn

%O 0,2

%A _Natan Arie Consigli_, Nov 29 2015