login
Square array read by rows: H_k(n,n), with rows n >= 0 and columns 0 <= k <= 3, where H_n is the n-th hyperoperation.
1

%I #12 Jan 11 2020 15:57:47

%S 1,0,0,1,2,2,1,1,3,4,4,4,4,6,9,27,5,8,16,256,6,10,25,3125,7,12,36,

%T 46656,8,14,49,823543,9,16,64,16777216,10,18,81,387420489,11,20,100,

%U 10000000000,12,22,121,285311670611,13,24,144,8916100448256,14,26,169,302875106592253

%N Square array read by rows: H_k(n,n), with rows n >= 0 and columns 0 <= k <= 3, where H_n is the n-th hyperoperation.

%C See A054871 for definitions and key links.

%C The purpose of this sequence is to unify all the visible terms of the sequence a(k)= H_k(n,n) for some n.

%e Square array begins:

%e 1, 0, 0, 1;

%e 2, 2, 1, 1;

%e 3, 4, 4, 4;

%e 4, 6, 9, 27;

%e 5, 8, 16, 256;

%e 6, 10, 25, 3125;

%e 7, 12, 36, 46656;

%e 8, 14, 49, 823543;

%e 9, 16, 64, 16777216;

%e 10, 18, 81, 387420489;

%e 11, 20, 100, 10000000000;

%e 12, 22, 121, 285311670611;

%e 13, 24, 144, 8916100448256;

%e 14, 26, 169, 302875106592253;

%e ...

%e For line n=10 we have:

%e H_0(10,10) = 11.

%e H_1(10,10) = 20.

%e H_2(10,10) = 100.

%e H_3(10,10) = 10000000000.

%t H[0, x_, y_] := y + 1;

%t H[1, x_, y_] := x + y;

%t H[2, x_, y_] := x*y;

%t H[3, x_, y_] := x^y;

%t H[4, x_, 2] := x^x;

%t Table[H[k, n, n], {n, 0, 20}, {k, 0, 3}]

%Y Cf. A054871, A280265 (table of H_k(n,2)).

%K nonn,tabf

%O 0,5

%A _Natan Arie Consigli_, Dec 30 2016