# Download American Mathematical Monthly, volume 117, August September by Daniel J. Velleman PDF

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.

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