Tetranacci Number
The tetranacci numbers are a generalization of the Fibonacci numbers defined by 
, 
, 
, 
, and the recurrence relation
 |
(1)
|
for 
. They represent the 
case of the
Fibonacci n-step numbers. The first
few terms for 
, 1, ... are 0, 1, 1, 2, 4, 8, 15,
29, 56, 108, 208, ... (OEIS A000078).
The first few prime tetranacci numbers have indices 3, 7, 11, 12, 36, 56, 401, 2707, 8417, 14096, 31561, 50696, 53192, 155182, ... (OEIS A104534),
corresponding to 2, 29, 401, 773, 5350220959, ... (OEIS A104535),
with no others for 
(E. W. Weisstein,
Mar. 21, 2009).
An exact expression for the 
th tetranacci number
for 
can be given explicitly by
 |
(2)
|
where the three additional terms are obtained by cyclically permuting 
,
which are the four roots of the polynomial
 |
(3)
|
Alternately,
 |
(4)
|
This can be written in slightly more concise form as
 |
(5)
|
where 
is the 
th root of the polynomial
 |
(6)
|
and 
and 
are in the ordering of the Wolfram Language's
Root object.
The tetranacci numbers have the generating function
 |
(7)
|
The ratio of adjacent terms tends to the positive real root of 
, namely 1.92756...
(OEIS A086088), which is sometimes known as
the tetranacci constant.
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tetranacci number