A Generalized Fibonacci Sequence

Some identities on conditional sequences by using matrix method

“…Horadam [3] has considered a generalized sequence fW n g defined by W n D pW n 1 qW n 2 ; n 2 with the initial conditions W 0 D a and W 1 D b, where a; b; p; q are arbitrary integers. If we take a D 0; b D 1 in fW n g; we get the generalized Fibonacci sequence and if we take a D 2; b D p in fW n g; we get the generalized Lucas sequence.…”

A New Generalization of Fibonacci Sequence & Extended Binet's Formula

“…The first few terms of the Fibonacci sequence are: 0, 1, 1, 2, 3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584, . That is, F n D F n 1 C F n 2 for all n 2.…”

“…The first few terms of the Fibonacci sequence are: 0, 1, 1, 2, 3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584, . Some authors [13,15,17,27,37] have generalized the Fibonacci sequence by preserving the recurrence relation and altering the first two terms of the sequence, while others [7,20,21,22,26,30,40] have generalized the Fibonacci sequence by preserving the first two terms of the sequence but altering the recurrence relation slightly. .…”

Binomial Transforms of the k-Fibonacci Sequence

“…The ratio of two consecutive Fibonacci numbers converges to the Golden Section, τ-, which appears in modern research [1][2][3][4][5], particularly Physics of the high energy particles [6,7] or theoretical Physics [8][9][10][11]. The ratio of two consecutive Fibonacci numbers converges to the Golden Section, τ-, which appears in modern research [1][2][3][4][5], particularly Physics of the high energy particles [6,7] or theoretical Physics [8][9][10][11].…”

“…Fibonacci numbers have been generalized in many ways [2,[15][16][17][18][19][20][21][22]. Gazale [20], Kappraff [21], and later Stakhov [22], Falcon and Plaza [18,19] independently introduced the Generalized Fibonacci numbers of the order k or simply Fibonacci Λτ-numbers, depending only on one integer parameter k.…”