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
ashimk:
20181209 18:40:07
Method Of Differences came into rescue.


ani1107:
20180924 08:42:18
Passed all test cases given in question but still it is showing wrong answer!


ani1107:
20180924 08:19:55
Why is Langrage's interpolation giving wrong answer?


fragile_dreams:
20180815 19:37:26
how to solve for 1 1 1 1 1 1 1 1 1 1 1 1 2 sequence??????? 

vaishcr7:
20180507 13:28:20
i guess 1 5 5 1 50 148 197 295 is an invalid test case because it can't be formed into a polynomial . can anyone tell me why my answer is failing after using differences method and clearing all the test cases of the problem that have already been provided. Last edit: 20180507 13:28:47 

heptapod:
20180112 15:08:04
never mind solved it!! for those who are having difficulty solving do go for 'Method of differences'....


heptapod:
20180112 12:31:46
Can some one please give me more testcases?? all the testcases mentioned here are showing correct output for my code but i am getting wrong answer still.. 

anurag_05:
20171214 19:02:33
Is this test case valid ; 1, 3, 9, 27, 81 

armageddon117:
20171130 17:38:27
Great Problem!! Last edit: 20171130 17:38:40 

simbha:
20170922 01:21:53
The solution works fine for O(n^2) time complexity! Feeling happy after One go AC! 
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 