Two Extensions of the Sury's Identity

Complementary Families of the Fibonacci-Lucas Relations

“…The next Theorem 4 gives an extension of the relation (7). It provides the answer whether there is an identity involving the product 3 n l n , respectively 3 n f n (which appears in some other contexts [4]).…”

“…It can be proved using Binet's formula [8], and by means of generating function as well [6]. Here we continue to call it Sury's identity, as it is already done in [6,7]. Lemma 3.…”

“…The same argument proves that m n L n represents the number of colored nbracelet tilings with squares in m colors and dominoes in m 2 colors. There are a few ways to prove Fibonacci-Lucas identity (7). It can be proved using Binet's formula [8], and by means of generating function as well [6].…”

An Alternating Sum of Fibonacci and Lucas Numbers of Order k

“…In the present note, we state and prove a new identity regarding an alternating sum of Fibonacci and Lucas numbers of order k, analogous to (2). As a special case of this identity for k = 2, a new identity follows that further reduces to a Fibonacci-Lucas relation derived recently by Martinjak [15].…”

Analogues of a Fibonacci-Lucas Identity

“…A recent proof of this identity by Sury [15] excited a lot of comment. Kwong [10] gave an alternate proof, and Marques [11] (see also Martinjak [12]) found an analogous identity that replaces 2 by 3. Marques' identity can be written (in the form presented by Martinjak):…”