Pell, Jacobsthal and other second-order sequences

Pell, Pell–Lucas, Jacobsthal, gibonacci, Horadam, Leonardo, Oresme, bi-periodic and (a,b)-Fibonacci sequences.

• (a, b)-Fibonacci and (a, b)-Lucas numbers
• Gibonacci numbers
• Horadam and generalized Fibonacci sequences
• Integer values of recurrence generating functions
• Binomial sums with second-order recurrences
• Integral representations of k-Jacobsthal numbers
• Jacobsthal and Jacobsthal–Lucas numbers
• Leonardo numbers
• Hyper-Leonardo numbers
• Normal second-order sequences
• Pell and Pell–Lucas numbers
• Brousseau sums of k-Fibonacci numbers
• Oresme numbers
• Toeplitz–Hessenberg determinants of recurrences

TheoremCassini's identity for associated Pell numbers#✓fully proved

Let \((Q_n)_{n \ge 0}\) be a sequence of integers with \(Q_0 = Q_1 = 1\) and \(Q_{n+2} = 2Q_{n+1} + Q_n\) for every \(n \ge 0\). Then for every natural number \(k \ge 1\),

\[Q_{k-1} Q_{k+1} = Q_k^2 + 2(-1)^{k-1}.\]

theorem associated_pell_cassini_identity
    (Q : ℕ → ℤ)
    (hQ_zero : Q 0 = 1)
    (hQ_one : Q 1 = 1)
    (hQ_rec : ∀ n : ℕ, Q (n + 2) = 2 * Q (n + 1) + Q n)
    (k : ℕ)
    (hk : 1 ≤ k) :
    Q (k - 1) * Q (k + 1) =
      Q k ^ 2 + 2 * (-1 : ℤ) ^ (k - 1)

Sources

• aBa Mbirika, Janee Schrader, and Jürgen Spilker, "Pell and Associated Pell Braid Sequences as GCDs of Sums of k Consecutive Pell, Balancing, and Related Numbers," Journal of Integer Sequences 26 (2023), Article 23.6.4, Lemma [Cassini's identity for Q].

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