T(n) = n(n+1)/2 are the well known triangular numbers.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Recursive Series Problem

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