-1,1

x[n]y = H_n(x,y) is the aggregation of x and y using the n-th hyperoperator. See A054871 for hyperoperator definitions and key links.

In a(-1) no hyperoperator is applied to 3 so a(-1) = 3. For n>0 calculate the chain a(n) = 3[0]3[1]3...3[n-1]3[n]3. Hyperoperators of higher degree have the higher aggregation priority, so tetration before exponentiation, exponentiation before multiplication, multiplication before addition, etc.

Wikipedia, Hyperoperation.

a(-1) = 3, a(n) = 3[0]3[1]3...3[n-1]3[n]3.

a(0) = 3[0]3 = '3 = 4;

a(1) = 3[0]3[1]3 = '3+3 = 7;

a(2) = 3[0]3[1]3[2]3 = '3+3*3 = 13;

a(3) = 3[0]3[1]3[2]3[3]3 = '3+3*3^3 = 85;

a(4) > 3^7625597484988 (a 3638334640025-digit number).

(PARI) f(x, y, o) = {if (o==4, z=x; for (i=1, y-1, z = x^z); return (z)); if (o==3, return(x^y)); if (o==2, return(x*y)); if (o==1, return(x+y)); }

a(n) = {t = 3; if (n>4, return("too big")); if (n==-1, return(t)); v = vector(n+1, k, t); w = vector(n+1, k, n-k+1); x = v[1]; for (k=1, n+1, if (w[k], x = f(v[k+1], x, w[k]), x = x+1); ); x; } \\ Michel Marcus, Jul 29 2015

Cf. A000012 (0[0]0[1]...[n]0), A157532 (1[0]1[1]...[n]1), A254225 (2[0]2[1]...[n]2).

Cf. A054871.

Sequence in context: A282718 A092406 A250297 * A121174 A050071 A041002

Adjacent sequences: A254307 A254308 A254309 * A254311 A254312 A254313

nonn,less

Natan Arie Consigli, May 03 2015

approved