


from collections import defaultdict
from math import ceil, sqrt






def  __lowerCAmelCase					(SCREAMING_SNAKE_CASE = 100_0000      ,      SCREAMING_SNAKE_CASE = 10      )->						int:




      """simple docstring"""

      snake_case_ =   defaultdict(SCREAMING_SNAKE_CASE      )

      for outer_width in range(3      ,      (t_limit // 4) + 2      ):
            if outer_width * outer_width > t_limit:
                  snake_case_ =   max(
                      ceil(sqrt(outer_width * outer_width - t_limit      )      )      ,      1      )
            else:
                  snake_case_ =   1

            hole_width_lower_bound += (outer_width - hole_width_lower_bound) % 2

            for hole_width in range(SCREAMING_SNAKE_CASE      ,      outer_width - 1      ,      2      ):
                  count[outer_width * outer_width - hole_width * hole_width] += 1

      return sum(1 for n in count.values() if 1 <= n <= 10      )


if __name__ == "__main__":
   print(f'''{solution() = }''')



import numpy as np

import datasets


UpperCAmelCase    =					"""
Compute the Mahalanobis Distance

Mahalonobis distance is the distance between a point and a distribution.
And not between two distinct points. It is effectively a multivariate equivalent of the Euclidean distance.
It was introduced by Prof. P. C. Mahalanobis in 1936
and has been used in various statistical applications ever since
[source: https://www.machinelearningplus.com/statistics/mahalanobis-distance/]
"""

UpperCAmelCase    =					"""\
@article{de2000mahalanobis,
  title={The mahalanobis distance},
  author={De Maesschalck, Roy and Jouan-Rimbaud, Delphine and Massart, D{\'e}sir{\'e} L},
  journal={Chemometrics and intelligent laboratory systems},
  volume={50},
  number={1},
  pages={1--18},
  year={2000},
  publisher={Elsevier}
}
"""

UpperCAmelCase    =					"""
Args:
    X: List of datapoints to be compared with the `reference_distribution`.
    reference_distribution: List of datapoints from the reference distribution we want to compare to.
Returns:
    mahalanobis: The Mahalonobis distance for each datapoint in `X`.
Examples:

    >>> mahalanobis_metric = datasets.load_metric(\"mahalanobis\")
    >>> results = mahalanobis_metric.compute(reference_distribution=[[0, 1], [1, 0]], X=[[0, 1]])
    >>> print(results)
    {'mahalanobis': array([0.5])}
"""







@datasets.utils.file_utils.add_start_docstrings(_DESCRIPTION      ,		_KWARGS_DESCRIPTION )
class lowerCAmelCase_		(				datasets.Metric ):

 '''simple docstring'''

 def 		UpperCamelCase__  (			self      ):
       return datasets.MetricInfo(
           description=_DESCRIPTION	,   citation=_CITATION	,   inputs_description=_KWARGS_DESCRIPTION	,   features=datasets.Features(
               {
                   '''X''': datasets.Sequence(datasets.Value('''float'''	,   id='''sequence'''      )	,   id='''X'''      ),
               }      )	,   )





 def 		UpperCamelCase__  (			self	,   _UpperCAmelCase	,   _UpperCAmelCase      ):
       # convert to numpy arrays
       snake_case_ =   np.array(_UpperCAmelCase      )
       snake_case_ =   np.array(_UpperCAmelCase      )

       # Assert that arrays are 2D
       if len(X.shape      ) != 2:
             raise ValueError('''Expected `X` to be a 2D vector'''      )
       if len(reference_distribution.shape      ) != 2:
             raise ValueError('''Expected `reference_distribution` to be a 2D vector'''      )
       if reference_distribution.shape[0] < 2:
             raise ValueError(
                 '''Expected `reference_distribution` to be a 2D vector with more than one element in the first dimension'''      )

       # Get mahalanobis distance for each prediction
       snake_case_ =   X - np.mean(_UpperCAmelCase      )
       snake_case_ =   np.cov(reference_distribution.T      )
       try:
             snake_case_ =   np.linalg.inv(_UpperCAmelCase      )
       except np.linalg.LinAlgError:
             snake_case_ =   np.linalg.pinv(_UpperCAmelCase      )
       snake_case_ =   np.dot(_UpperCAmelCase	,   _UpperCAmelCase      )
       snake_case_ =   np.dot(_UpperCAmelCase	,   X_minus_mu.T      ).diagonal()

       return {"mahalanobis": mahal_dist}