The purpose of the present paper is to prove that there are finitely many binomial coefficients of the form (f in certain binary recurrences, and give a simple method for the determination of these coefficients. We illustrate the method by the Fibonacci, the Lucas, and the Pell sequences. First, we transform both of the title equations into two elliptic equations and apply a theorem of Mordell ...

Let {ai,j} be real numbers for 0 6 i 6 r 1 and 1 6 j 6 2, and define a sequence {vn} with initial conditions v0, v1 and conditional linear recurrence relation vn = at,1vn 1 + at,2vn 2 where n ⌘ t (mod r) (n > 2). The sequence {vn} can be viewed as a generalization of many well-known integer sequences, such as Fibonacci, Lucas, Pell, Jacobsthal, etc. We find explicitly a linear recurrence equati...

We provide tiling proofs of several algebraic formulas for the Pell numbers of odd index, all of which involve alternating sums of binomial coefficients, as well as consider polynomial generalizations of these formulas. In addition, we provide a combinatorial interpretation for a Diophantine equation satisfied by the Pell numbers of odd index.

The second-order linear recurrence sequence (wn(a, b; p, q))n≥0, or briefly (wn)n≥0, is defined by wn+2 = pwn+1 + qwn, (1) with w0 = a, w1 = b and n ≥ 0. This sequence was introduced in 1965 by Horadam [3, 4], and it generalizes many sequences (see [1, 5]). Examples of such sequences are Fibonacci number sequences (Fn)n≥0, Lucas number sequences (Ln)n≥0, and Pell number sequences (Pn)n≥0, when ...

is (an(c0, . . . , cl−1; p1, . . . , pl))n≥0, or briefly (an)n≥0, where the pi are constant coefficients, with given aj = cj for all j = 0, 1, . . . , l − 1, and n ≥ l; in such a context, (an)n≥0 is called an l-sequence. In the case l = 2, this sequence is called Horadam’s sequence and was introduced, in 1965, by Horadam [4, 5], and it generalizes many sequences (see [1, 6]). Examples of such s...

We investigate some arithmetic properties of the q-Fibonacci numbers and the q-Pell numbers.

In this paper, the matrices related to Fibonacci, Lucas, Pell, and Pell-Lucas numbers have been examined. By using these new identities integer sequences investigated.