ulman-corner-explanation.groovy

// https://oeis.org/A054554
// https://www.youtube.com/watch?v=iFuR97YcSLM
for(int x = 2; x < 10; x++) {
  //4n^2-2n+1
  def N = 4 * Math.pow(x, 2) - 2 * x + 1
  //ᒪ√((n+√(n-4)) / 4)ᒧ
  def U = Math.floor(Math.sqrt((N + Math.sqrt(N - 4)) / 4));
  assert x == U
  def u = U+1 // +1 para que se vea igual a https://oeis.org/A054554/b054554.txt
  println String.format("%.0f -> %.0f", u, N)
}

//Comprobación
for(int x = 2; x < 9999; x++) {
  def N = 4 * Math.pow(x, 2) - 2 * x + 1
  assert N == 4 * Math.pow(x, 2) - 2 * x + 1
  assert N - 1 == 4 * Math.pow(x, 2) - 2 * x
  assert N - 1 == 2 * (2 * Math.pow(x, 2) - x)
  assert (N - 1) / 2 == 2 * Math.pow(x, 2) - x
  assert (N - 1) / 2 == x * (2 * x - 1)
  //No se una mejor forma de despejar (-1), así que se busca su equivaliente en función de N; x=f(N)
  assert (N - 1) / 2 != x * (2 * x)
  assert ((N - 1) / 2) + (Math.sqrt(N)) != x * (2 * x)
  assert ((N - 1) / 2) + (N + Math.sqrt(N)) != x * (2 * x)
  assert ((N - 1) / 2) + ((N + Math.sqrt(N)) / 4) != x * (2 * x)
  assert ((N - 1) / 2) + Math.sqrt((N + Math.sqrt(N)) / 4) != x * (2 * x)
  assert ((N - 1) / 2) + Math.floor(Math.sqrt((N + Math.sqrt(N)) / 4)) == x * (2 * x)
  assert ((N - 1) / 2) + Math.floor(Math.sqrt((N + Math.sqrt(N)) / 4)) == x * 2 * x
  assert ((N - 1) / 2) + Math.floor(Math.sqrt((N + Math.sqrt(N)) / 4)) == Math.pow(x, 2) * 2
  assert (((N - 1) / 2) + Math.floor(Math.sqrt((N + Math.sqrt(N)) / 4))) / 2 == Math.pow(x, 2)
  assert Math.sqrt((((N - 1) / 2) + Math.floor(Math.sqrt((N + Math.sqrt(N)) / 4))) / 2) == x
  //La versión más corta que tengo
  assert Math.floor(Math.sqrt((N + Math.sqrt(N - 4)) / 4)) == x
}