The Fibonacci sequence is: \[1,1,2,3,5,8,13,\cdots\]
The $n^{th}$ term is usually denoted $F_n$.
We use L = [1,1,2,3,5,8,13] to specify a list (zero-indexed sequence) and stores it to L.
One can access the $1^{st}$ term with L[1].
L = [1,1,2,3,5,8,13]
In programming, list/sequence are usually indexed from zero.
It is common to extend the sequence by the zero-th term, i.e. when $n=0$.
Recall that $$F_{n+2} = F_{n+1} + F_n $$, by setting $n=0$, we have $F_n = 0$.
F = [0,1,1,2,3,5,8,13]
Thus, we can align the index in sage as usual sequence.
To access a single element of a list, we use the square bracket to enclose the index.
Index can be negative (as in modular arithmetic).
F = [0,1,1,2,3,5,8,13]
F[6] # This is 8
F[-1] # The rightmost element
F[-2] # 8 again
We can use the slice format [start:stop:step] to take a sub-list.
The range is specified with inclusive open and exclusive ending.
F = [0,1,1,2,3,5,8,13]
F[1:] # A sub-list with elements of F[1], F[2], ... (i.e. dropping the 0-th term)
F[:-1] # A sub-list without the last element
F[1:3] # A list of just two elements of F[1], F[2]
F[::2] # A list [0,1,3,8]
To specify a list of consecutive integers, e.g. [ 3 .. 100 ].
range(100) can be used to specify a list of 100 consecutive integers starting from 0.
To construct a list of 10 square numbers: [k^2 for k in range(10)].
To construct a list of 10 even numbers: [2*k for k in range(10)].
A list of numbers within 100 that is not divisible by 3:
[ k for k in range(100) if k%3 != 0 ]
A list of numbers withint 1000 that is divisible by 2 or 3:
[ k for k in range(1000) if k%2 == 0 or k%3 == 0 ]
== is used for equality comparison
!= is used for comparison, i.e. $\neq$.
| Name | Description | Example |
|---|---|---|
sum |
Total sum | sum([1..100]) |
prod |
Total product | prod([1..10]) |
len |
Total length of list | len(range(10)) |
count |
Number of occurences | [1,1,2,3,1,1,2,1,1].count(1) |