text: """
  PROBLEM
  
  If we are presented with the first k terms of a sequence it is impossible to say with certainty the value
  of the next term, as there are infinitely many polynomial functions that can model the sequence.
  
  As an example, let us consider the sequence of cube numbers. This is defined by the generating function,
  u_n = n^3: 1, 8, 27, 64, 125, 216, ...
  
  Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best"
  we should assume a linear relationship and predict the next term to be 15 (common difference 7). Even if we were
  presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.
  
  We shall define OP(k, n) to be the nth term of the optimum polynomial generating function for the first k terms of
  a sequence. It should be clear that OP(k, n) will accurately generate the terms of the sequence for n ≤ k, and
  potentially the first incorrect term (FIT) will be OP(k, k+1); in which case we shall call it a bad OP (BOP).
  
  As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy;
  that is, for n ≥ 2, OP(1, n) = u1.
  
  Hence we obtain the following OPs for the cubic sequence:
  
  OP(1, n) = 1      	 1, <1>, 1, 1, ...
  OP(2, n) = 7n−6	     1, 8, <15>, ...
  OP(3, n) = 6n^2−11n+6  1, 8, 27, <58>, ...
  OP(4, n) = n^3	     1, 8, 27, 64, 125, ...
  Clearly no BOPs exist for k ≥ 4.
  
  By considering the sum of FITs generated by the BOPs (indicated in <> above), we obtain 1 + 15 + 58 = 74.
  
  Consider the following tenth degree polynomial generating function:
  
  u_n = 1 − n + n^2 − n^3 + n^4 − n^5 + n^6 − n^7 + n^8 − n^9 + n^10
  
  Find the sum of FITs for the BOPs.
  
  ANSWER: 37076114526
  Solve time ~ 0.098 seconds
  """
  import numpy as np
  import unittest
  from util.utils import timeit
  
  
  class Problem101:
      def __init__(self, max_power):
          self.max_power = max_power
          self.ans = 0
  
      def generating_poly(self, n):
          return (1 + n ** (self.max_power + 1)) / (1 + n)
  
      @timeit
      def solve(self):
          for k in range(1, self.max_power + 1):
              ls_coef = np.polyfit(y=[self.generating_poly(i) for i in range(1, k + 1)], x=range(1, k + 1), deg=k - 1)
              self.ans += int(np.poly1d(ls_coef)(k + 1) + 0.5)
  
          return self.ans
  
      def get_solution(self):
          return self.ans
  
  
  class Solution101(unittest.TestCase):
      def setUp(self):
          self.problem = Problem101(max_power=10)
  
      def test_solution(self):
          self.assertEqual(37076114526, self.problem.solve())
  
  
  if __name__ == '__main__':
      unittest.main()
  