Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Recursive Series Problem

  1. May 28, 2005 #1
    T(n) = n(n+1)/2 are the well known triangular numbers.
    Let A(0) = 2 let A(1) = 3 A(n) = 6*A(n-1) -A(n-2) - 4. This series
    gives 2, 3, 12, 65 etc. The product of any two adjacent terms is
    always a triangular number. For instance 2*3 = T(3); 3*12 = T(8),
    12*65 = T(39) etc.
    In fact A(0) can be any integer and still there would always be an
    infinite number of solutions in integers for A(1) and N such that the
    recursive series defined by A(0), A(1) and A(n) = 6*A(n-1) - A(n-2) -
    has the property that the product of two adjacent terms is always a
    triangular number.
    The problem is to find at least one solution for A(1) and N for each
    of A(0) = 1, 4, 5, 6, 7, 8 and 9 respectfully. A solution for A(0) =
    2 or 3 is the example given above.
    Bonus: explain a rule, formula or workable method for finding
    solutions for any given A(0) that does not simply involve trial and
    error or computer searching.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted