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CTK Exchange
Ramsey_KJ
Member since Sep-23-04
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Nov-13-04, 10:38 AM (EST) |
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"Recursive Series"
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This problem appears in the yahoo groups IntermediateNumberTheory Group. The following concerns the set of recursive series: R_0 = 0, R_1 = 2^ (n+1)-1 and R_i = 6*R_(i-1) - R_(i-2) + K where K = 4*(2^n -1) (for n = 0, K = 0 and (R_i)^2 is a triangular number). The product any two adjacent terms of these series is also a triangular number (can be represented as the product x*(x+1)/2 where x is an integer) if n is an positive interger (0,1,2,3 etc.). Otherwise the product can still be put in the form x*(x+1)/2 where x_i are integers times a negative power of two. A special case is K = -4 (limit as n -> minus infinity) in that case the R series (where the product of any two adjacent terms is a triangular number) is 5, 2, 3, 12, 65 etc, and x_i are also integers.Challenge 1. Find a non-recursive formula for the ith term of the R series for a given K (other than 0). 2. Find the recursive series x_i for the triangular numbers x_i * (x_i+1)/2 represented by R_i*R_(i-1). Hint it is of the form 6* x_(i- 1) - x_(i-2) + K2 where K2 is a constant dependent on K. 3. Find a non recursive formula for the ith term x_i corresponding to a given K. Have a Good Day KJ Ramsey Have a Good Day KJ Ramsey
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