.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/combinatorics/sequences/plot_01_fibonacci_rabbits.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_combinatorics_sequences_plot_01_fibonacci_rabbits.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_combinatorics_sequences_plot_01_fibonacci_rabbits.py:


Fibonacci's rabbits and domino tilings
============================================

Reproduces the rabbit problem from Fibonacci's 1202 *Liber Abaci*,
checks that the same numbers count domino tilings of a 2-by-n strip,
and shows the ratio of successive terms approaching the golden ratio.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.combinatorics import domino_tilings, fibonacci








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Rabbit pairs month by month
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-25

.. code-block:: Python


    # Liber Abaci starts from one pair and counts the pairs after each month.
    print("month:       " + " ".join(f"{m:4d}" for m in range(0, 13)))
    print("rabbit pairs:" + " ".join(f"{fibonacci(m + 2):4d}" for m in range(0, 13)))
    print(f"after a year: {fibonacci(14)} pairs, Fibonacci's own answer")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    month:          0    1    2    3    4    5    6    7    8    9   10   11   12
    rabbit pairs:   1    2    3    5    8   13   21   34   55   89  144  233  377
    after a year: 377 pairs, Fibonacci's own answer




.. GENERATED FROM PYTHON SOURCE LINES 26-28

Tilings of a 2-by-n strip
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-32

.. code-block:: Python


    for n in range(1, 8):
        print(f"2 x {n} strip: {domino_tilings(n)} tilings")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    2 x 1 strip: 1 tilings
    2 x 2 strip: 2 tilings
    2 x 3 strip: 3 tilings
    2 x 4 strip: 5 tilings
    2 x 5 strip: 8 tilings
    2 x 6 strip: 13 tilings
    2 x 7 strip: 21 tilings




.. GENERATED FROM PYTHON SOURCE LINES 33-35

Ratios converge to the golden ratio
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-44

.. code-block:: Python


    n = np.arange(2, 30)
    ratios = [fibonacci(k + 1) / fibonacci(k) for k in n]
    phi = (1 + np.sqrt(5)) / 2
    fig, ax = plt.subplots()
    ax.semilogy(n, [abs(r - phi) for r in ratios], "o-")
    ax.set_xlabel("n")
    ax.set_ylabel(r"$|F_{n+1}/F_n - \varphi|$")
    ax.set_title("Successive ratios approach the golden ratio")



.. image-sg:: /api/gallery/combinatorics/sequences/images/sphx_glr_plot_01_fibonacci_rabbits_001.png
   :alt: Successive ratios approach the golden ratio
   :srcset: /api/gallery/combinatorics/sequences/images/sphx_glr_plot_01_fibonacci_rabbits_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'Successive ratios approach the golden ratio')




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.031 seconds)


.. _sphx_glr_download_api_gallery_combinatorics_sequences_plot_01_fibonacci_rabbits.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_fibonacci_rabbits.ipynb <plot_01_fibonacci_rabbits.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_fibonacci_rabbits.py <plot_01_fibonacci_rabbits.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_fibonacci_rabbits.zip <plot_01_fibonacci_rabbits.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_