Pentanacci Number
المؤلف:
Sloane, N. J. A
المصدر:
Sequence A001591/M1122 and A103814 in "The On-Line Encyclopedia of Integer Sequences
الجزء والصفحة:
...
7-12-2020
1206
Pentanacci Number
The pentanacci 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 
, 2, ... are 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ... (OEIS A001591).
The ratio of adjacent terms tends to the real root of 
, namely 1.965948236645485... (OEIS A103814), sometimes called the pentanacci constant.
An exact formula for the 
th pentanacci number can be given explicitly in terms of the five roots 
of
 |
(2)
|
as
 |
(3)
|
The pentanacci numbers have generating function
 |
(4)
|
REFERENCES:
Sloane, N. J. A. Sequence A001591/M1122 and A103814 in "The On-Line Encyclopedia of Integer Sequences."
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