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import FormalConjecturesUtilCombinatorial representation of the recurrence $a(n+2) = 2a(n+1) + 2a(n)$
A028859 (OEIS): $a(n+2) = 2 \cdot a(n+1) + 2 \cdot a(n)$; $a(0) = 1$, $a(1) = 3$.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA28859A028859 (OEIS): $a(n+2) = 2 \cdot a(n+1) + 2 \cdot a(n)$; $a(0) = 1$, $a(1) = 3$.
def a (n : ℕ) : ℕ :=
match n with
| 0 => 1
| 1 => 3
| (n + 2) => 2 * a (n + 1) + 2 * a n
termination_by nset_option linter.unusedVariables false@[category test, AMS 11]
lemma a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_1 : a 1 = 3 := ⊢ a 1 = 3 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 8 := ⊢ a 2 = 8 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 22 := ⊢ a 3 = 22 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 60 := ⊢ a 4 = 60 All goals completed! 🐙Conjecture: The sequence a(n) is also the number of compositions of $n$ into positive integers such that adjacent parts and the largest part differ by at most 1. - Gus Wiseman, May 19 2020
A formal proof has been found with the methods described in arxiv/2605.22763.
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/28859.wip.lean#L402"]
theorem exists_finset_sequence (n : ℕ) :
let L := n + 1
let Sequence := Fin L → ℕ
let S : Set Sequence :=
{σ : Sequence |
L > 0 ∧
(∀ i : Fin L, σ i > 0) ∧
let max_val := Finset.sup Finset.univ σ
(∀ k : ℕ, 1 ≤ k ∧ k ≤ max_val → ∃ i : Fin L, σ i = k) ∧
(∀ i j : Fin L, i < j → j.val ≠ i.val + 1 → σ i ≥ σ j)}
∃ (F : Finset Sequence), (F : Set Sequence) = S ∧ F.card = a n := n:ℕ⊢ let L := n + 1;
let Sequence := Fin L → ℕ;
let S :=
{σ |
L > 0 ∧
(∀ (i : Fin L), σ i > 0) ∧
let max_val := Finset.univ.sup σ;
(∀ (k : ℕ), 1 ≤ k ∧ k ≤ max_val → ∃ i, σ i = k) ∧ ∀ (i j : Fin L), i < j → ↑j ≠ ↑i + 1 → σ i ≥ σ j};
∃ F, ↑F = S ∧ F.card = a n
All goals completed! 🐙end OeisA28859