Fibonacci series in reverse order
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. The Fibonacci series is a classic mathematical concept that has many applications in various fields, including computer science and algorithms.
In this context, we are interested in generating the Fibonacci series in reverse order using a recursive
algorithm. Given a positive integer n
, we want to find the last n
Fibonacci numbers in
reverse order.
Explanation using Example
Let's consider the two test cases from the provided code:
Test A: n = 10
The first 10 Fibonacci numbers are [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
. When displayed in reverse
order, they become [34, 21, 13, 8, 5, 3, 2, 1, 1, 0]
, which is the expected output.
Test B: n = 15
The first 15 Fibonacci numbers are [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]
. When
displayed in reverse order, they become [377, 233, 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0]
,
which is the expected output.
Pseudocode
fibonacciNumber(a, b, n)
if n <= 0
return
fibonacciNumber(b, a + b, n  1)
print a
Algorithm Explanation
The fibonacciNumber
function is a recursive algorithm to generate the last n
Fibonacci
numbers in reverse order. It takes three parameters: a
, b
, and n
. The
variables a
and b
represent the two preceding Fibonacci numbers, and n
represents the count of remaining Fibonacci numbers to be generated.
The algorithm starts by checking if n
is less than or equal to 0. If n
is nonpositive,
it returns, as there are no more Fibonacci numbers to generate.
Next, it recursively calls fibonacciNumber
with the parameters b
and a + b
(the next Fibonacci number), and decrements n
by 1. This step ensures that the algorithm generates
the Fibonacci numbers in reverse order.
Finally, the algorithm prints the current value of a
, which represents the next Fibonacci number in
reverse order.
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Time Complexity
The time complexity of the provided algorithm is O(n)
, where n
is the number of
Fibonacci numbers to generate. This is because the algorithm makes n
recursive calls, and each call
takes constant time to compute the next Fibonacci number and print it. Hence, the overall time complexity is
linear with respect to the input n
.
Resultant Output Explanation
For Test A with n = 10
, the algorithm generates the last 10 Fibonacci numbers in reverse order, and
the output is [34, 21, 13, 8, 5, 3, 2, 1, 1, 0]
, which matches the expected result.
For Test B with n = 15
, the algorithm generates the last 15 Fibonacci numbers in reverse order, and
the output is [377, 233, 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0]
, which also matches the
expected result.
The code successfully generates the Fibonacci series in reverse order using recursion for the given test cases.
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