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By Daniel J. Velleman

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We have to show that ti 2k−1 + 2k , which is equivalent to ti 2k−1 + 2k ≤ 614 c √ 2 ti 2k−2 + γi 2k−li −2 + 1 2 < ti 2k−1 + 2k + 1, THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 or in other words, √ √ √ √ 2 2 βi + γi − 2li +1 + 2 αi 2 2k−li −2 0≤2 βi − 2 2 √ √ 2 < 1. − αi 2 2k−li −1 + 2 k−li −1 Since γi − βi = 2αi and αi + βi = 2li +1 this is the same as √ √ √ 0 ≤ {αi 2 2k−li −1 } − 2 {αi 2 2k−li −2 } + √ 2 < 1. 2 (7) √ √ Relation (7) is true since 0 ≤ {x} − 2 {x/2} + 2/2 < √ 1 for all x ∈ R.

J. A. Sloane’s online encyclopedia of integer sequences [12] gives eight sequences which are connected to the Graham–Pollak sequence (1), namely, A091522, A091523, A091524, A091525, A100671, A100673, A001521, and A004539. Recently [13, 14], the present author found vast extensions of the Graham–Pollak sequence to parametric families of recurrences, where the initial value u 1 = 1 is re√ placed by u 1 = m and the 2 in the recurrence is accordingly changed. However, √ the sequence is still wrapped in considerable mystery.

Baney, S. Beslin, and V. DeAngelis, Farey tree and distribution of small denominators, Topology Proc. 22 (1997) 23–37. 3. M. Bicknell-Johnson, Stern diatomic array applied to Fibonacci representations, Fibonacci Quart. 41 (2003) 169–180. , The Fibonacci diatomic array applied to Fibonacci representations, in Applications of Fibonacci 4. Numbers, vol. 9, Kluwer Academic, Dordrecht, 2004, 29–38. 5. N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly 107 (2000) 360–367. 2307/2589182 6.

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