





def 					_A	(		SCREAMING_SNAKE_CASE					:       int						):





  """simple docstring"""



  a__     : Any  =(1 + 24 * n) ** 0.5
  return ((1 + root) / 6) % 1 == 0
def 					_A	(		SCREAMING_SNAKE_CASE					:       int = 5_000						):





  """simple docstring"""



  a__     : Optional[int]  =[(i * (3 * i - 1)) // 2 for i in range(1	,	SCREAMING_SNAKE_CASE						)]
  for i, pentagonal_i in enumerate(SCREAMING_SNAKE_CASE						):
    for j in range(SCREAMING_SNAKE_CASE	,	len(SCREAMING_SNAKE_CASE						)						):
      a__     : List[Any]  =pentagonal_nums[j]
      a__     : Optional[Any]  =pentagonal_i + pentagonal_j
      a__     : Optional[Any]  =pentagonal_j - pentagonal_i
      if is_pentagonal(SCREAMING_SNAKE_CASE						) and is_pentagonal(SCREAMING_SNAKE_CASE						):
        return b

  return -1


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













from math import acos, sin
from typing import List, Tuple, Union

import numpy as np
import torch
from PIL import Image

from ...models import AutoencoderKL, UNetaDConditionModel
from ...schedulers import DDIMScheduler, DDPMScheduler
from ...utils import randn_tensor
from ..pipeline_utils import AudioPipelineOutput, BaseOutput, DiffusionPipeline, ImagePipelineOutput
from .mel import Mel







class     __lowerCAmelCase	(   UpperCamelCase__):
      _lowercase    : str            =      ["""vqvae"""]







      def __init__(   self		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				)       ->						Union[str, Any]:



        '''simple docstring'''




        super().__init__()
        self.register_modules(unet=lowerCAmelCase__		,				scheduler=lowerCAmelCase__		,				mel=lowerCAmelCase__		,				vqvae=lowerCAmelCase__							)







      def        _lowercase  (   self							)       ->						int:



        '''simple docstring'''




        return 5_0 if isinstance(self.scheduler		,				lowerCAmelCase__							) else 1_0_0_0







      @torch.no_grad()
      def __call__(   self		,				lowerCAmelCase__ = 1		,				lowerCAmelCase__ = None		,				lowerCAmelCase__ = None		,				lowerCAmelCase__ = 0		,				lowerCAmelCase__ = 0		,				lowerCAmelCase__ = None		,				lowerCAmelCase__ = None		,				lowerCAmelCase__ = 0		,				lowerCAmelCase__ = 0		,				lowerCAmelCase__ = None		,				lowerCAmelCase__ = 0		,				lowerCAmelCase__ = None		,				lowerCAmelCase__ = None		,				lowerCAmelCase__=True		,				)       ->						Union[
          Union[AudioPipelineOutput, ImagePipelineOutput],
          Tuple[List[Image.Image], Tuple[int, List[np.ndarray]]],
      ]:



        '''simple docstring'''




        a__     : List[Any]  =steps or self.get_default_steps()
        self.scheduler.set_timesteps(lowerCAmelCase__							)
        a__     : Tuple  =step_generator or generator
        # For backwards compatibility
        if type(self.unet.config.sample_size							) == int:
          a__     : List[str]  =(self.unet.config.sample_size, self.unet.config.sample_size)
        if noise is None:
          a__     : Optional[Any]  =randn_tensor(
              (
                  batch_size,
                  self.unet.config.in_channels,
                  self.unet.config.sample_size[0],
                  self.unet.config.sample_size[1],
              )		,				generator=lowerCAmelCase__		,				device=self.device		,				)
        a__     : List[str]  =noise
        a__     : Optional[Any]  =None

        if audio_file is not None or raw_audio is not None:
          self.mel.load_audio(lowerCAmelCase__		,				lowerCAmelCase__							)
          a__     : Tuple  =self.mel.audio_slice_to_image(lowerCAmelCase__							)
          a__     : List[Any]  =np.frombuffer(input_image.tobytes()		,				dtype="uint8"							).reshape(
              (input_image.height, input_image.width)							)
          a__     : Optional[Any]  =(input_image / 2_5_5) * 2 - 1
          a__     : Dict  =torch.tensor(input_image[np.newaxis, :, :]		,				dtype=torch.float							).to(self.device							)

          if self.vqvae is not None:
            a__     : str  =self.vqvae.encode(torch.unsqueeze(lowerCAmelCase__		,				0							)							).latent_dist.sample(
                generator=lowerCAmelCase__							)[0]
            a__     : Any  =self.vqvae.config.scaling_factor * input_images

          if start_step > 0:
            a__     : Optional[int]  =self.scheduler.add_noise(lowerCAmelCase__		,				lowerCAmelCase__		,				self.scheduler.timesteps[start_step - 1]							)

          a__     : Tuple  =(
              self.unet.config.sample_size[1] * self.mel.get_sample_rate() / self.mel.x_res / self.mel.hop_length
          )
          a__     : Union[str, Any]  =int(mask_start_secs * pixels_per_second							)
          a__     : List[str]  =int(mask_end_secs * pixels_per_second							)
          a__     : Optional[Any]  =self.scheduler.add_noise(lowerCAmelCase__		,				lowerCAmelCase__		,				torch.tensor(self.scheduler.timesteps[start_step:]							)							)

        for step, t in enumerate(self.progress_bar(self.scheduler.timesteps[start_step:]							)							):
          if isinstance(self.unet		,				lowerCAmelCase__							):
            a__     : List[str]  =self.unet(lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__							)["sample"]
          else:
            a__     : Optional[Any]  =self.unet(lowerCAmelCase__		,				lowerCAmelCase__							)["sample"]

          if isinstance(self.scheduler		,				lowerCAmelCase__							):
            a__     : int  =self.scheduler.step(
                model_output=lowerCAmelCase__		,				timestep=lowerCAmelCase__		,				sample=lowerCAmelCase__		,				eta=lowerCAmelCase__		,				generator=lowerCAmelCase__		,				)["prev_sample"]
          else:
            a__     : str  =self.scheduler.step(
                model_output=lowerCAmelCase__		,				timestep=lowerCAmelCase__		,				sample=lowerCAmelCase__		,				generator=lowerCAmelCase__		,				)["prev_sample"]

          if mask is not None:
            if mask_start > 0:
              a__     : List[Any]  =mask[:, step, :, :mask_start]
            if mask_end > 0:
              a__     : Union[str, Any]  =mask[:, step, :, -mask_end:]

        if self.vqvae is not None:
          # 0.18215 was scaling factor used in training to ensure unit variance
          a__     : Any  =1 / self.vqvae.config.scaling_factor * images
          a__     : str  =self.vqvae.decode(lowerCAmelCase__							)["sample"]

        a__     : str  =(images / 2 + 0.5).clamp(0		,				1							)
        a__     : int  =images.cpu().permute(0		,				2		,				3		,				1							).numpy()
        a__     : List[Any]  =(images * 2_5_5).round().astype("uint8"							)
        a__     : Dict  =list(
            (Image.fromarray(_[:, :, 0]							) for _ in images)
            if images.shape[3] == 1
            else (Image.fromarray(lowerCAmelCase__		,				mode="RGB"							).convert("L"							) for _ in images)							)

        a__     : str  =[self.mel.image_to_audio(lowerCAmelCase__							) for _ in images]
        if not return_dict:
          return images, (self.mel.get_sample_rate(), audios)

        return BaseOutput(**AudioPipelineOutput(np.array(lowerCAmelCase__							)[:, np.newaxis, :]							)		,				**ImagePipelineOutput(lowerCAmelCase__							)							)







      @torch.no_grad()
      def        _lowercase  (   self		,				lowerCAmelCase__		,				lowerCAmelCase__ = 5_0							)       ->						np.ndarray:



        '''simple docstring'''




        assert isinstance(self.scheduler		,				lowerCAmelCase__							)
        self.scheduler.set_timesteps(lowerCAmelCase__							)
        a__     : Union[str, Any]  =np.array(
            [np.frombuffer(image.tobytes()		,				dtype="uint8"							).reshape((1, image.height, image.width)							) for image in images]							)
        a__     : Tuple  =(sample / 2_5_5) * 2 - 1
        a__     : List[Any]  =torch.Tensor(lowerCAmelCase__							).to(self.device							)

        for t in self.progress_bar(torch.flip(self.scheduler.timesteps		,				(0,)							)							):
          a__     : str  =t - self.scheduler.config.num_train_timesteps // self.scheduler.num_inference_steps
          a__     : Dict  =self.scheduler.alphas_cumprod[t]
          a__     : Optional[Any]  =(
              self.scheduler.alphas_cumprod[prev_timestep]
              if prev_timestep >= 0
              else self.scheduler.final_alpha_cumprod
          )
          a__     : Optional[Any]  =1 - alpha_prod_t
          a__     : str  =self.unet(lowerCAmelCase__		,				lowerCAmelCase__							)["sample"]
          a__     : Optional[Any]  =(1 - alpha_prod_t_prev) ** 0.5 * model_output
          a__     : List[str]  =(sample - pred_sample_direction) * alpha_prod_t_prev ** (-0.5)
          a__     : Optional[Any]  =sample * alpha_prod_t ** 0.5 + beta_prod_t ** 0.5 * model_output

        return sample







      @staticmethod
      def        _lowercase  (   lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__							)       ->						torch.Tensor:



        '''simple docstring'''




        a__     : Any  =acos(torch.dot(torch.flatten(lowerCAmelCase__							)		,				torch.flatten(lowerCAmelCase__							)							) / torch.norm(lowerCAmelCase__							) / torch.norm(lowerCAmelCase__							)							)
        return sin((1 - alpha) * theta							) * xa / sin(lowerCAmelCase__							) + sin(alpha * theta							) * xa / sin(lowerCAmelCase__							)






