Figure out this algorithm by analyzing the sequence of numbers generated

Question

You are given a function f that, using an algorithm (that uses an old method), performs a calculation on four distinct values _ and generates a sequence of ten numbers.

Now, your task is to figure out how the algorithm works based on the outputs received for a set of inputs.

Inputs and Outputs:

f([1, 1, 1, 1])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([2, 2, 2, 2])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([3, 3, 3, 3])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([4, 4, 4, 4])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([5, 5, 5, 5])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([6, 6, 6, 6])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([8, 8, 8, 8])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([7, 7, 7, 7])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([9, 9, 9, 9])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([1, 2, 3, 6])=[5, 0, 1, 2, 3, 4, 5, 0, 1, 2]
f([2, 3, 6, 5])=[2, 4, 1, 3, 0, 2, 4, 1, 3, 0]
f([4, 1, 7, 9])=[2, 6, 1, 5, 0, 4, 8, 3, 7, 2]
f([5, 2, 6, 7])=[2, 0, 5, 3, 1, 6, 4, 2, 0, 5]
f([8, 1, 3, 2])=[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
f([4, 6, 5, 8])=[5, 1, 5, 1, 5, 1, 5, 1, 5, 1]
f([1, 7, 9, 44])=[16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
f([1, 6, 47, 2])=[1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
f([45, 87, 95, 68])=[66, 43, 20, 65, 42, 19, 64, 41, 18, 63]
f([335, 66, 48, 97])=[42, 86, 33, 77, 24, 68, 15, 59, 6, 50]
f([48, 796, 548, 1])=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
f([8794, 35, 3, 3])=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2]
f([89, 554, 665, 7])=[5, 3, 1, 6, 4, 2, 0, 5, 3, 1]
f([54, 89, 32, 67])=[14, 1, 55, 42, 29, 16, 3, 57, 44, 31]
f([21, 65, 78, 49])=[22, 43, 15, 36, 8, 29, 1, 22, 43, 15]
f([3, 5, 11, 9])=[8, 2, 5, 8, 2, 5, 8, 2, 5, 8]
f([37, 42, 13, 128])=[31, 68, 105, 14, 51, 88, 125, 34, 71, 108]
f([48, 76, 2, 34])=[12, 26, 6, 20, 0, 14, 28, 8, 22, 2]
f([94, 57, 62, 111])=[92, 75, 58, 41, 24, 7, 101, 84, 67, 50]
f([111, 222, 333, 664])=[407, 518, 629, 76, 187, 298, 409, 520, 631, 78]
f([795, 1, 45, 3333])=[840, 1635, 2430, 3225, 687, 1482, 2277, 3072, 534, 1329]
f([897, 4, 6, 2])=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
f([68, 749, 44, 88])=[24, 4, 72, 52, 32, 12, 80, 60, 40, 20]
f([78946, 6656, 4, 10])=[0, 6, 2, 8, 4, 0, 6, 2, 8, 4]
f([8, 1, 222, 77])=[76, 7, 15, 23, 31, 39, 47, 55, 63, 71]
f([1, 2, 3, 99999])=[5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

Answer 1

Let's call $A,B,C,D$ the parameters of the function. The first term of the sequence is

$((A\times B)+C)$mod $D$

Each consequent term is obtained from the previous this way:

Add $A$ then modulo $D$

The $n$th term of the sequence can be expressed as:

$(A\times (B+n)+C)mod\ D$

If you need a proof of the equivalence of the two definitions, just ask in the comments.

Example:

f([89, 554, 665, 7])=[5, 3, 1, 6, 4, 2, 0, 5, 3, 1]

The first term is

$(89\times 554+665) mod\ 7 =5$

The second is

$(5+89)mod\ 7=3$

And the $n$th is

$(89\times (554+n)+665)mod\ 7$

Answer 2

This function delivers

an arithmetic sequence.

The first element of the parameter is

the number to add on each iteration.

The second element of the parameter is

the initial value of our iterator.

The third element of the parameter is

the initial offset of the sequence.

The fourth element of the parameter is

the modulus.

The function is described in code as:

def f(x):
  r = []
  for i in range(x[1],x[1]+10):
    r += [(x[2] + (i*x[0]))%x[3]]
  return r