maths.special_numbers.jacobsthal_number

Jacobsthal numbers

The Jacobsthal sequence starts with 0 and 1, and every later term is the previous term plus twice the term before it:

J(0) = 0 J(1) = 1 J(n) = J(n - 1) + 2 * J(n - 2)

The first terms are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, …

They also have the closed form J(n) = (2 ** n - (-1) ** n) / 3, which is used below to cross-check the iterative result.

Source:

https://en.wikipedia.org/wiki/Jacobsthal_number https://oeis.org/A001045

Functions

jacobsthal_number(→ int)	Return the nth Jacobsthal number, counting from J(0) = 0.
jacobsthal_sequence(→ list[int])	Return the first length Jacobsthal numbers as a list.

Module Contents

maths.special_numbers.jacobsthal_number.jacobsthal_number(n: int) → int

Return the nth Jacobsthal number, counting from J(0) = 0.

Parameters:

n – index of the Jacobsthal number, must be a non-negative integer

Returns:

the nth Jacobsthal number

>>> jacobsthal_number(0)
0
>>> jacobsthal_number(1)
1
>>> jacobsthal_number(5)
11
>>> jacobsthal_number(10)
341
>>> jacobsthal_number(64)
6148914691236517205
>>> all(
...     jacobsthal_number(i) == (2**i - (-1) ** i) // 3 for i in range(100)
... )
True
>>> jacobsthal_number(-1)
Traceback (most recent call last):
    ...
ValueError: n must be a non-negative integer, got -1
>>> jacobsthal_number(2.5)
Traceback (most recent call last):
    ...
ValueError: n must be a non-negative integer, got 2.5
>>> jacobsthal_number("3")
Traceback (most recent call last):
    ...
ValueError: n must be a non-negative integer, got '3'

maths.special_numbers.jacobsthal_number.jacobsthal_sequence(length: int) → list[int]

Return the first length Jacobsthal numbers as a list.

Parameters:

length – how many terms to return, must be a non-negative integer

Returns:

list of the first length Jacobsthal numbers

>>> jacobsthal_sequence(0)
[]
>>> jacobsthal_sequence(1)
[0]
>>> jacobsthal_sequence(11)
[0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341]
>>> jacobsthal_sequence(-3)
Traceback (most recent call last):
    ...
ValueError: length must be a non-negative integer, got -3
>>> jacobsthal_sequence(4.0)
Traceback (most recent call last):
    ...
ValueError: length must be a non-negative integer, got 4.0