Theory and Modern Applications
Table 1 Special cases of the sequence \(\{w_{n}\}\)
From: Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences
\(\{w_{n}\}\) | \(\{w_{n} ( w_{0},w_{1};a,b,c ) \}\) | generalized bi-periodic Horadam sequence |
|---|---|---|
\(\{u_{n}\}\) | \(\{w_{n} ( 0,1;a,b,c ) \}\) | generalized bi-periodic Fibonacci sequence |
\(\{v_{n}\}\) | \(\{w_{n} ( 2,b;a,b,c ) \}\) | generalized bi-periodic Lucas sequence |
\(\{q_{n}\}\) | \(\{w_{n} ( 0,1;a,b,1 ) \}\) | bi-periodic Fibonacci sequence [4] |
\(\{p_{n}\}\) | \(\{w_{n} ( 2,a;b,a,1 ) \}\) | bi-periodic Lucas sequence [2] |
\(\{W_{n}\}\) | \(\{w_{n} ( w_{0},w_{1};a,b,1 ) \}\) | bi-periodic Horadam sequence [4] |
\(\{H_{n}\}\) | \(\{w_{n} ( w_{0},w_{1};p,p,-q ) \}\) | Horadam sequence [5] |
\(\{F_{n}\}\) | \(\{w_{n} ( 0,1;1,1,1 ) \}\) | Fibonacci sequence |
\(\{L_{n}\}\) | \(\{w_{n} ( 2,1;1,1,1 ) \}\) | Lucas sequence |
\(\{F_{k,n}\}\) | \(\{w_{n} ( 0,1;k,k,1 ) \}\) | k-Fibonacci sequence |
\(\{L_{k,n}\}\) | \(\{w_{n} ( 0,k;k,k,1 ) \}\) | k-Lucas sequence |
\(\{P_{n}\}\) | \(\{w_{n} ( 0,1;2,2,1 ) \}\) | Pell sequence |
\(\{PL_{n}\}\) | \(\{w_{n} ( 2,2;2,2,1 ) \}\) | Pell–Lucas sequence |
\(\{J_{n}\}\) | \(\{w_{n} ( 0,1;1,1,2 ) \}\) | Jacobsthal sequence |
\(\{JL_{n}\}\) | \(\{w_{n} ( 2,1;1,1,2 ) \}\) | Jacobsthal–Lucas sequence |