






from math import ceil, sqrt

def   __lowercase       (			snake_case = 1_0_0_0_0_0_0			):







				"""simple docstring"""






				__magic_name__		:Any   =   0

				for outer_width in range(3,	(limit // 4) + 2			):
								if outer_width**2 > limit:
												__magic_name__		:Any   =   max(ceil(sqrt(outer_width**2 - limit			)			),	1			)
								else:
												__magic_name__		:List[str]   =   1
								if (outer_width - hole_width_lower_bound) % 2:
												hole_width_lower_bound += 1

								answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

				return answer


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














import unittest

import numpy as np

from transformers.testing_utils import require_flax, require_tf, require_torch
from transformers.utils import (
    expand_dims,
    flatten_dict,
    is_flax_available,
    is_tf_available,
    is_torch_available,
    reshape,
    squeeze,
    transpose,
)


if is_flax_available():
					import jax.numpy as jnp

if is_tf_available():
					import tensorflow as tf

if is_torch_available():
					import torch
class     lowerCamelCase_					(   unittest.TestCase			):







				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:List[Any]   =   {
								    '''task_specific_params''': {
								        '''summarization''': {'''length_penalty''': 1.0, '''max_length''': 1_2_8, '''min_length''': 1_2, '''num_beams''': 4},
								        '''summarization_cnn''': {'''length_penalty''': 2.0, '''max_length''': 1_4_2, '''min_length''': 5_6, '''num_beams''': 4},
								        '''summarization_xsum''': {'''length_penalty''': 1.0, '''max_length''': 6_2, '''min_length''': 1_1, '''num_beams''': 6},
								    }
								}
								__magic_name__		:List[str]   =   {
								    '''task_specific_params.summarization.length_penalty''': 1.0,
								    '''task_specific_params.summarization.max_length''': 1_2_8,
								    '''task_specific_params.summarization.min_length''': 1_2,
								    '''task_specific_params.summarization.num_beams''': 4,
								    '''task_specific_params.summarization_cnn.length_penalty''': 2.0,
								    '''task_specific_params.summarization_cnn.max_length''': 1_4_2,
								    '''task_specific_params.summarization_cnn.min_length''': 5_6,
								    '''task_specific_params.summarization_cnn.num_beams''': 4,
								    '''task_specific_params.summarization_xsum.length_penalty''': 1.0,
								    '''task_specific_params.summarization_xsum.max_length''': 6_2,
								    '''task_specific_params.summarization_xsum.min_length''': 1_1,
								    '''task_specific_params.summarization_xsum.num_beams''': 6,
								}

								self.assertEqual(flatten_dict(__lowerCAmelCase  )							,       __lowerCAmelCase  )







				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Optional[Any]   =   np.random.randn(3							,       4  )
								self.assertTrue(np.allclose(transpose(__lowerCAmelCase  )							,       x.transpose()  )  )

								__magic_name__		:List[Any]   =   np.random.randn(3							,       4							,       5  )
								self.assertTrue(np.allclose(transpose(__lowerCAmelCase							,       axes=(1, 2, 0)  )							,       x.transpose((1, 2, 0)  )  )  )







				@require_torch
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Optional[Any]   =   np.random.randn(3							,       4  )
								__magic_name__		:Tuple   =   torch.tensor(__lowerCAmelCase  )
								self.assertTrue(np.allclose(transpose(__lowerCAmelCase  )							,       transpose(__lowerCAmelCase  ).numpy()  )  )

								__magic_name__		:int   =   np.random.randn(3							,       4							,       5  )
								__magic_name__		:Union[str, Any]   =   torch.tensor(__lowerCAmelCase  )
								self.assertTrue(np.allclose(transpose(__lowerCAmelCase							,       axes=(1, 2, 0)  )							,       transpose(__lowerCAmelCase							,       axes=(1, 2, 0)  ).numpy()  )  )







				@require_tf
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:int   =   np.random.randn(3							,       4  )
								__magic_name__		:Optional[Any]   =   tf.constant(__lowerCAmelCase  )
								self.assertTrue(np.allclose(transpose(__lowerCAmelCase  )							,       transpose(__lowerCAmelCase  ).numpy()  )  )

								__magic_name__		:List[str]   =   np.random.randn(3							,       4							,       5  )
								__magic_name__		:Optional[Any]   =   tf.constant(__lowerCAmelCase  )
								self.assertTrue(np.allclose(transpose(__lowerCAmelCase							,       axes=(1, 2, 0)  )							,       transpose(__lowerCAmelCase							,       axes=(1, 2, 0)  ).numpy()  )  )







				@require_flax
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:int   =   np.random.randn(3							,       4  )
								__magic_name__		:Dict   =   jnp.array(__lowerCAmelCase  )
								self.assertTrue(np.allclose(transpose(__lowerCAmelCase  )							,       np.asarray(transpose(__lowerCAmelCase  )  )  )  )

								__magic_name__		:Dict   =   np.random.randn(3							,       4							,       5  )
								__magic_name__		:Dict   =   jnp.array(__lowerCAmelCase  )
								self.assertTrue(np.allclose(transpose(__lowerCAmelCase							,       axes=(1, 2, 0)  )							,       np.asarray(transpose(__lowerCAmelCase							,       axes=(1, 2, 0)  )  )  )  )







				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Any   =   np.random.randn(3							,       4  )
								self.assertTrue(np.allclose(reshape(__lowerCAmelCase							,       (4, 3)  )							,       np.reshape(__lowerCAmelCase							,       (4, 3)  )  )  )

								__magic_name__		:Union[str, Any]   =   np.random.randn(3							,       4							,       5  )
								self.assertTrue(np.allclose(reshape(__lowerCAmelCase							,       (1_2, 5)  )							,       np.reshape(__lowerCAmelCase							,       (1_2, 5)  )  )  )







				@require_torch
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Dict   =   np.random.randn(3							,       4  )
								__magic_name__		:Tuple   =   torch.tensor(__lowerCAmelCase  )
								self.assertTrue(np.allclose(reshape(__lowerCAmelCase							,       (4, 3)  )							,       reshape(__lowerCAmelCase							,       (4, 3)  ).numpy()  )  )

								__magic_name__		:Union[str, Any]   =   np.random.randn(3							,       4							,       5  )
								__magic_name__		:List[str]   =   torch.tensor(__lowerCAmelCase  )
								self.assertTrue(np.allclose(reshape(__lowerCAmelCase							,       (1_2, 5)  )							,       reshape(__lowerCAmelCase							,       (1_2, 5)  ).numpy()  )  )







				@require_tf
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Dict   =   np.random.randn(3							,       4  )
								__magic_name__		:Union[str, Any]   =   tf.constant(__lowerCAmelCase  )
								self.assertTrue(np.allclose(reshape(__lowerCAmelCase							,       (4, 3)  )							,       reshape(__lowerCAmelCase							,       (4, 3)  ).numpy()  )  )

								__magic_name__		:List[Any]   =   np.random.randn(3							,       4							,       5  )
								__magic_name__		:Optional[int]   =   tf.constant(__lowerCAmelCase  )
								self.assertTrue(np.allclose(reshape(__lowerCAmelCase							,       (1_2, 5)  )							,       reshape(__lowerCAmelCase							,       (1_2, 5)  ).numpy()  )  )







				@require_flax
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:List[str]   =   np.random.randn(3							,       4  )
								__magic_name__		:Any   =   jnp.array(__lowerCAmelCase  )
								self.assertTrue(np.allclose(reshape(__lowerCAmelCase							,       (4, 3)  )							,       np.asarray(reshape(__lowerCAmelCase							,       (4, 3)  )  )  )  )

								__magic_name__		:List[Any]   =   np.random.randn(3							,       4							,       5  )
								__magic_name__		:List[str]   =   jnp.array(__lowerCAmelCase  )
								self.assertTrue(np.allclose(reshape(__lowerCAmelCase							,       (1_2, 5)  )							,       np.asarray(reshape(__lowerCAmelCase							,       (1_2, 5)  )  )  )  )







				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:List[Any]   =   np.random.randn(1							,       3							,       4  )
								self.assertTrue(np.allclose(squeeze(__lowerCAmelCase  )							,       np.squeeze(__lowerCAmelCase  )  )  )

								__magic_name__		:Optional[Any]   =   np.random.randn(1							,       4							,       1							,       5  )
								self.assertTrue(np.allclose(squeeze(__lowerCAmelCase							,       axis=2  )							,       np.squeeze(__lowerCAmelCase							,       axis=2  )  )  )







				@require_torch
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Dict   =   np.random.randn(1							,       3							,       4  )
								__magic_name__		:List[Any]   =   torch.tensor(__lowerCAmelCase  )
								self.assertTrue(np.allclose(squeeze(__lowerCAmelCase  )							,       squeeze(__lowerCAmelCase  ).numpy()  )  )

								__magic_name__		:List[str]   =   np.random.randn(1							,       4							,       1							,       5  )
								__magic_name__		:str   =   torch.tensor(__lowerCAmelCase  )
								self.assertTrue(np.allclose(squeeze(__lowerCAmelCase							,       axis=2  )							,       squeeze(__lowerCAmelCase							,       axis=2  ).numpy()  )  )







				@require_tf
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:int   =   np.random.randn(1							,       3							,       4  )
								__magic_name__		:Tuple   =   tf.constant(__lowerCAmelCase  )
								self.assertTrue(np.allclose(squeeze(__lowerCAmelCase  )							,       squeeze(__lowerCAmelCase  ).numpy()  )  )

								__magic_name__		:Tuple   =   np.random.randn(1							,       4							,       1							,       5  )
								__magic_name__		:Optional[int]   =   tf.constant(__lowerCAmelCase  )
								self.assertTrue(np.allclose(squeeze(__lowerCAmelCase							,       axis=2  )							,       squeeze(__lowerCAmelCase							,       axis=2  ).numpy()  )  )







				@require_flax
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Tuple   =   np.random.randn(1							,       3							,       4  )
								__magic_name__		:Optional[Any]   =   jnp.array(__lowerCAmelCase  )
								self.assertTrue(np.allclose(squeeze(__lowerCAmelCase  )							,       np.asarray(squeeze(__lowerCAmelCase  )  )  )  )

								__magic_name__		:List[Any]   =   np.random.randn(1							,       4							,       1							,       5  )
								__magic_name__		:Optional[Any]   =   jnp.array(__lowerCAmelCase  )
								self.assertTrue(np.allclose(squeeze(__lowerCAmelCase							,       axis=2  )							,       np.asarray(squeeze(__lowerCAmelCase							,       axis=2  )  )  )  )







				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Any   =   np.random.randn(3							,       4  )
								self.assertTrue(np.allclose(expand_dims(__lowerCAmelCase							,       axis=1  )							,       np.expand_dims(__lowerCAmelCase							,       axis=1  )  )  )







				@require_torch
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:List[Any]   =   np.random.randn(3							,       4  )
								__magic_name__		:Any   =   torch.tensor(__lowerCAmelCase  )
								self.assertTrue(np.allclose(expand_dims(__lowerCAmelCase							,       axis=1  )							,       expand_dims(__lowerCAmelCase							,       axis=1  ).numpy()  )  )







				@require_tf
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:Union[str, Any]   =   np.random.randn(3							,       4  )
								__magic_name__		:Union[str, Any]   =   tf.constant(__lowerCAmelCase  )
								self.assertTrue(np.allclose(expand_dims(__lowerCAmelCase							,       axis=1  )							,       expand_dims(__lowerCAmelCase							,       axis=1  ).numpy()  )  )







				@require_flax
				def       A						(    self  ):
								"""simple docstring"""
								__magic_name__		:List[str]   =   np.random.randn(3							,       4  )
								__magic_name__		:Tuple   =   jnp.array(__lowerCAmelCase  )
								self.assertTrue(np.allclose(expand_dims(__lowerCAmelCase							,       axis=1  )							,       np.asarray(expand_dims(__lowerCAmelCase							,       axis=1  )  )  )  )






