code: def _initial_nodes_a(n, k):
      r"""Tricomi initial guesses
  
      Computes an initial approximation to the square of the `k`-th
      (positive) root :math:`x_k` of the Hermite polynomial :math:`H_n`
      of order :math:`n`. The formula is the one from lemma 3.1 in the
      original paper. The guesses are accurate except in the region
      near :math:`\sqrt{2n + 1}`.
  
      Parameters
      ----------
      n : int
          Quadrature order
      k : ndarray of type int
          Index of roots to compute
  
      Returns
      -------
      xksq : ndarray
          Square of the approximate roots
  
      See Also
      --------
      initial_nodes
      roots_hermite_asy
      """
      tauk = _compute_tauk(n, k)
      sigk = cos(0.5 * tauk) ** 2
      a = n % 2 - 0.5
      nu = 4.0 * floor(n / 2.0) + 2.0 * a + 2.0
      # Initial approximation of Hermite roots (square)
      xksq = nu * sigk - 1.0 / (3.0 * nu) * (
          5.0 / (4.0 * (1.0 - sigk) ** 2) - 1.0 / (1.0 - sigk) - 0.25
      )
      return xksq
  
imports: import numpy
  from numpy import exp, inf, pi, sqrt, floor, sin, cos, around, hstack, arccos, arange
  from scipy import linalg
  from scipy.special import airy
  from .None import _specfun
  from .None import _ufuncs
label_desc: Use image node_id 12 for calling a global function with example usage: _initial_nodes_a(n, k) and returns: xksq