An step Fibonacci sequence is given by defining for , , , and

(1) 
for . The case corresponds to the degenerate 1, 1, 2, 2, 2, 2 ..., to the usual Fibonacci
Numbers 1, 1, 2, 3, 5, 8, ... (Sloane's A000045), to the Tribonacci Numbers 1, 1, 2, 4, 7, 13, 24, 44, 81, ... (Sloane's A000073), to the Tetranacci Numbers
1, 1, 2, 4, 8, 15, 29, 56, 108, ... (Sloane's A000078), etc.
The limit
is given by solving

(2) 
for and taking the Real Root . If , the equation reduces to

(3) 

(4) 
giving solutions

(5) 
The ratio is therefore

(6) 
which is the Golden Ratio, as expected.
Solutions for , 2, ... are given numerically by 1, 1.61803, 1.83929, 1.92756, 1.96595, ..., approaching 2 as
.
See also Fibonacci Number, Tribonacci Number
References
Sloane, N. J. A. Sequences
A000045/M0692,
A000073/M1074, and
A000078/M1108
in ``An OnLine Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
© 19969 Eric W. Weisstein
19990526