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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) -
    N
    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
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