1,3

This sequence grows faster than any exponential sequence. The implementation here is quite slow.

Let g(x) be fib(1,1,x), g returns y; let h(y) be fib(1,y,x), h returns z; let i(z) be z be applied to itself x-1 times. Then f(x) = i(h(g(x)))

E.g. n=1 fib(1,1,n) n=2 fib(1,fib(1,1,n),n) n=3 fib(1,fib(1,fib(1,1,n),n),n) ...

f(3) is fib(1,fib(1,fib(1,1,3),3),3)

f(3) simplifies to fib(1,fib(1,2,3),3)

f(3) simplifies to fib(1,3,3)

f(3) is 4

.Python program (replace leading dots by spaces):

.#This is Python, use slowfibnd(x) to generate the xth term.

.from string import *

.def fib(arb1, arb2, nth):

.....if nth == 0:

.........return arb1

.....if nth == 1:

.........return arb2

.....x = [0]*nth

.....x[0] = arb1

.....x[1] = arb2

.....for i in xrange(2, nth, 1):

.........x[i] = x[i-1]+x[i-2]

.....return x[-1]

.def fib2d(n):

.....return fib(1, fib(1, 1, n), n)

.def fib3d(n):

.....return fib(1, fib(1, fib(1, 1, n), n), n)

.def slowfibnd(n): #This is an inelegant solution, but it will work

.....begin = "fib(1, 0+1, n)"

.....for x in range(n-1):

.........begin = replace(begin, '0+1', 'fib(1, 0+1, n)')

.....return eval(begin)

Cf. A000045 is the Fibonacci function fib(1, 1, n), A142975 is the Fibonacci function applied to itself fib(1, fib(1, 1, n), n).

Sequence in context: A145087 A005046 A141827 this_sequence A005841 A005828 A084764

Adjacent sequences: A143074 A143075 A143076 this_sequence A143078 A143079 A143080

nonn

Gregory Nisbet (gregory.nisbet(AT)gmail.com), Jul 22 2008