Continuing the work of Zemke, Livingston and Allen, we consider when linear combinations of torus knots are concordant to L-space knots. We begin by proving Allen's conjecture for alternating torus knots. That is, we prove that a linear combination of alternating torus knots is concordant to an L-space knot if and only if the connected sum is a single torus knot. Then we establish a necessary condition for when a linear combination of torus knots is concordant to an L-space knot.

We explore the sums of k consecutive terms in the generalized Fibonacci sequence {G n } ∞ n=0 given by the recurrence G n = G n−1 + G n−2 for all n ≥ 2 with integral initial conditions G 0 and G 1 . In particular, we give precise values for the greatest common divisor (GCD) of all sums of k consecutive terms of {G n } ∞ n=0 . When G 0 = 0 and G 1 = 1, we yield the GCD of all sums of k consecutive Fibonacci numbers, and when G 0 = 2 and G 1 = 1, we yield the GCD of all sums of k consecutive Lucas numbers. Denoting the GCD of all sums of k consecutive generalized Fibonacci numbers by the symbol G G0,G1 (k), we give two tantalizing characterizations for these values, one involving a simple formula in k and another involving generalized Pisano periods:where π G0,G1 (m) denotes the generalized Pisano period of the generalized Fibonacci sequence modulo m. The fact that these vastly different-looking formulas coincide leads to some surprising and delightful new understandings of the Fibonacci and Lucas numbers.

The Fibonacci sequence modulo m, which we denote (F m,n ) ∞ n=0 where F m,n is the Fibonacci number F n modulo m, has been a well-studied object in mathematics since the seminal paper by D. D. Wall in 1960 exploring a myriad of properties related to the periods of these sequences. Since the time of Lagrange it has been known that (F m,n ) ∞ n=0 is periodic for each m. We examine this sequence when m = 10, yielding a sequence of period length 60. In particular, we explore its subsequences composed of every r th term of (F 10,n ) ∞ n=0 starting from the term F 10,k for some 0 ≤ k ≤ 59. More precisely we consider the subsequences (F 10,k+rj ) ∞ j=0 , which we show are themselves periodic and whose lengths divide 60. Many intriguing properties reveal themselves as we alter the k and r values. For example, for certain r values the corresponding subsequences surprisingly obey the Fibonacci recurrence relation; that is, any two consecutive subsequence terms sum to the next term modulo 10. Moreover, for all r values relatively prime to 60, the subsequence (F 10,k+rj ) ∞ j=0 coincides exactly with the original parent sequence (F 10,n ) ∞ n=0 (or a cyclic shift of that) running either forward or reverse. We demystify this phenomena and explore many other tantalizing properties of these subsequences.