A175796 H_n(n, 2) where H_c(a, b) is the hyperoperation function with operator c.

3, 3, 4, 9, 256

Offset: 0

Author

Grant Garcia, Sep 06 2010

Keywords

• nonn

Examples

			a(0) = H_0(0, 2) = 2 + 1 = 3
a(1) = H_1(1, 2) = 1 + 2 = 3
a(2) = H_2(2, 2) = 2 * 2 = 4
a(3) = H_3(3, 2) = 3 ^ 2 = 9
a(4) = H_4(4, 2) = 4 ^^ 2 = 4 ^ 4 = 256
a(5) = H_5(5, 2) = 5 ^^^ 2 = 5 ^^ 5 = 5 ^ 5 ^ 5 ^ 5 ^ 5 =~ 10 ^ (10 ^ (10 ^ (2184.1257...)))
		

Links

• Wikipedia, Hyperoperation
• Index Section Ho-Hy

Crossrefs

For H_n(x,2) with fixed x, cf. A054871 (x=3), A253855 (x=4), A255176 (x=2), A256131 (x=10), A261143 (x=1). - Danny Rorabaugh, Oct 20 2015

Programs

Python

def H(a, b, c):
    if c == 0: return b + 1
    if c == 1 and b == 0: return a
    if c == 2 and b == 0: return 0
    if c >= 3 and b == 0: return 1
    return H(a, H(a, b - 1, c), c - 1)
for n in range(5): print(H(n, 2, n))

Formula

a(n) = H_n(n, 2)

H_c(a, b) = {b + 1 if c = 0; a if c = 1, b = 0; 0 if c = 2, b = 0; 1 if c >= 3, b = 0; H_{c-1}(a, H_c(a, b - 1)) otherwise}