Abstract
In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.
Highlights
Nicomachus’ theorem asserts that the sum of the first m cubes is the square of the mth triangular number, 13 + 23 + · · · + m3 = (1 + 2 + · · · + m)2. (1.1)(See [2].) With the notation Q(α, m) := (where α ∈ R \ {0}, it implies that m n=1 m n=1 αn αn, )2 (1.2) lim Q(α, m) = α. m→∞ (1.3)Here, x is the integer part of the real number x
Is expressible as a fraction whose numerator and denominator are polynomials in Fibonacci and Lucas numbers with indices depending linearly on k according to the parity of k, yet the statement of Theorem 2.1 presents formulas for these quantities according to the congruence class of k modulo 4 rather than modulo 2
The discrepancy is related to different factorisations of the factors Fn − 1 that occur in the formulas for A(k, j) and A (k, j) for j ∈ {1, 3}, since each of the factors Fn − 1 happens to be a product of a Fibonacci and a Lucas number according to the congruence class of n modulo 4 (see formulas (6.1))
Summary
The corresponding result is Theorem 2.1 below. We complement it by a general analysis of moments of the Beatty sequences and give a solution to a related arithmetic question in Theorem 2.5
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