CMPLS  Complete the Sequence!
You probably know those quizzes in Sunday magazines: given the sequence 1, 2, 3, 4, 5, what is the next number? Sometimes it is very easy to answer, sometimes it could be pretty hard. Because these "sequence problems" are very popular, ACM wants to implement them into the "Free Time" section of their new WAP portal.
ACM programmers have noticed that some of the quizzes can be solved by describing the sequence by polynomials. For example, the sequence 1, 2, 3, 4, 5 can be easily understood as a trivial polynomial. The next number is 6. But even more complex sequences, like 1, 2, 4, 7, 11, can be described by a polynomial. In this case, 1/2.n^{2}1/2.n+1 can be used. Note that even if the members of the sequence are integers, polynomial coefficients may be any real numbers.
Polynomial is an expression in the following form:
If a_{D} <> 0, the number D is called a degree of the polynomial. Note that constant function P(n) = C can be considered as polynomial of degree 0, and the zero function P(n) = 0 is usually defined to have degree 1.
Input
There is a single positive integer T on the first line of input (equal to about 5000). It stands for the number of test cases to follow. Each test case consists of two lines. First line of each test case contains two integer numbers S and C separated by a single space, 1 <= S < 100, 1 <= C < 100, (S+C) <= 100. The first number, S, stands for the length of the given sequence, the second number, C is the amount of numbers you are to find to complete the sequence.
The second line of each test case contains S integer numbers X_{1}, X_{2}, ... X_{S} separated by a space. These numbers form the given sequence. The sequence can always be described by a polynomial P(n) such that for every i, X_{i} = P(i). Among these polynomials, we can find the polynomial P_{min} with the lowest possible degree. This polynomial should be used for completing the sequence.
Output
For every test case, your program must print a single line containing C integer numbers, separated by a space. These numbers are the values completing the sequence according to the polynomial of the lowest possible degree. In other words, you are to print values P_{min}(S+1), P_{min}(S+2), .... P_{min}(S+C).
It is guaranteed that the results P_{min}(S+i) will be nonnegative and will fit into the standard integer type.
Example
Sample Input:
4 6 3 1 2 3 4 5 6 8 2 1 2 4 7 11 16 22 29 10 2 1 1 1 1 1 1 1 1 1 2 1 10 3
Sample Output:
7 8 9 37 46 11 56 3 3 3 3 3 3 3 3 3 3Warning: large Input/Output data, be careful with certain languages
hide comments
drunkenstein:
20220220 21:14:18
AC in one go!!


raman_95:
20211025 10:36:50
Few patterns are understandable , but others cant understand . Can anyone provide any links to understand the problem with more clarity?


sabkx:
20210921 16:06:08
Lagrange interpolation formula should work here right? 

guru_shreyansh:
20210804 15:38:55
Solved it in Java using the Difference Method. Many people shared the wiki link for the Diff Method but that was not useful to me at all. You can refer to: https://brilliant.org/wiki/methodofdifferences/


king_rohan12:
20200721 16:25:12
i got the problem but can anyone explain me about


jsthiago:
20200503 22:26:15
i think the 1 1 1 1 1 1 1 1 1 2 case is when you do the diff table you got


rahul_t_7:
20200321 13:14:40
check this test case 1 1 1 1 1 1 1 Last edit: 20200321 13:15:30 

ngtrhieu:
20200129 12:18:46
Hmm I think Lanrange poly only give a polynomial that guarantee to fit at the input points, but does not guarantee other points.. 

sumit_989692:
20191224 08:07:59
Can someone explain me this testcase?1 1 1 1 1 1 1 1 1 2 

alok_0366:
20190610 14:56:15
what does this ques wants to say?? i understood the method of difference. 
Added by:  adrian 
Date:  20040508 
Time limit:  5s 
Source limit:  50000B 
Memory limit:  1536MB 
Cluster:  Cube (Intel G860) 
Languages:  All except: NODEJS PERL6 VB.NET 
Resource:  ACM Central European Programming Contest, Prague 2000 