Generalizations of third-order recurrence relation

Abstract

This paper presents a generalization of the sequence defined by the third-order recurrence relation𝑉𝑛 (𝑎 𝑗 , 𝑝 𝑗) = Í 3 𝑗=1 𝑝 𝑗𝑉𝑛− 𝑗 , 𝑛 ≥ 4,, 𝑝3 ≠ 0 with initial terms 𝑉𝑗 = 𝑎 𝑗 , where 𝑎 𝑗 and 𝑝 𝑗 𝑗 = 1, 2, 3, are any non-zero real numbers. The generating function and Binet’s formula are derived for this generalized tribonacci sequence. Classical second-order generalized Fibonacci sequences and other existing sequences based on second-order recurrence relations are implicitly included in this analysis. These derived sequences are discussed as special cases of the generalization. A pictorial representation is provided, illustrating the growth and variation of tribonacci numbers for different initial terms 𝑎 𝑗 and coefficients 𝑝 𝑗 . Additionally, the tribonacci constant is examined and visually represented. It is observed that the constant is influenced solely by the coefficients 𝑝 𝑗 of the recurrence relation and is unaffected by the initial terms 𝑎 𝑗 .

Keywords

• Generalized Tribonacci sequence
• generating function
• Binet’s formula
• Tribonacci constan

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