yaworsw/euler-manager

---
:id: 422
:name: Sequence of points on a hyperbola
:url: https://projecteuler.net/problem=422
:content: "Let H be the hyperbola defined by the equation 12<var>x</var><sup>2</sup>
  + 7<var>x</var><var>y</var> - 12<var>y</var><sup>2</sup> = 625.\n\nNext, define
  X as the point (7, 1). It can be seen that X is in H.\n\nNow we define a sequence
  of points in H, {P<sub><var>i</var></sub> : <var>i</var> ≥ 1}, as:\n\n- P<sub>1</sub>
  = (13, 61/4).\n- P<sub>2</sub> = (-43/6, -4).\n- For <var>i</var> \\> 2, P<sub><var>i</var></sub>
  is the unique point in H that is different from P<sub><var>i</var>-1</sub> and such
  that line P<sub><var>i</var></sub>P<sub><var>i</var>-1</sub> is parallel to line
  P<sub><var>i</var>-2</sub>X. It can be shown that P<sub><var>i</var></sub> is well-defined,
  and that its coordinates are always rational.\n ![p422_hyperbola.gif]({{ images_dir
  }}/p422_hyperbola.gif)\n\nYou are given that P<sub>3</sub> = (-19/2, -229/24), P<sub>4</sub>
  = (1267/144, -37/12) and P<sub>7</sub> = (17194218091/143327232, 274748766781/1719926784).\n\nFind
  P<sub><var>n</var></sub> for <var>n</var> = 11<sup>14</sup> in the following format:
  \ \nIf P<sub><var>n</var></sub> = (<var>a</var>/<var>b</var>, <var>c</var>/<var>d</var>)
  where the fractions are in lowest terms and the denominators are positive, then
  the answer is (<var>a</var> + <var>b</var> + <var>c</var> + <var>d</var>) mod 1&nbsp;000&nbsp;000&nbsp;007.\n\nFor
  <var>n</var> = 7, the answer would have been: 806236837.\n\n"