$a(n) = \sum_{i=0}^n \binom{2n-i}{n+i}$

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import FormalConjecturesUtil

/-!
# $a(n) = \sum_{i=0}^n \binom{2n-i}{n+i}$

Alternatively the sequence `a` can be defined as
$a(n) = \sum_{k=0}^n \binom{n+k-1}{k} F(n-k+1)$, where $F(m)$ is the $m$-th Fibonacci number.
We formalize a conjecture about the number of words of length $n$ in a set $X$ being related
to this sequence.

*References:*
- [A108081](https://oeis.org/A108081)
-/

namespace OeisA108081

open Nat

/-- The primary defining sequence `a`.
$a(n) = \sum_{k=0}^n \binom{n+k-1}{k} F(n-k+1)$, where $F(m)$ is the $m$-th Fibonacci number. -/
def a (n : ℕ) : ℕ :=
  ∑ k ∈ Finset.range (n + 1), (n + k - 1).choose k * fib (n - k + 1)

/-- Term theorems verifying the first few values of the sequence against the official OEIS b-file -/
@[category test, AMS 5]
theorem a_0 : a 0 = 1 := by decide

@[category test, AMS 5]
theorem a_1 : a 1 = 2 := by decide

@[category test, AMS 5]
theorem a_2 : a 2 = 7 := by decide

@[category test, AMS 5]
theorem a_3 : a 3 = 25 := by decide

@[category test, AMS 5]
theorem a_4 : a 4 = 92 := by decide

/-- Words are finite sequences of integers. -/
abbrev Word := List ℤ

/-- `l w` is the word obtained by reversing `w` and subtracting 1 from every term. -/
def l (w : Word) : Word :=
  w.reverse.map (fun x => x - 1)

/-- `r w` is the word obtained by reversing `w` and adding 1 to every term. -/
def r (w : Word) : Word :=
  w.reverse.map (fun x => x + 1)

/--
`XWord` is the smallest set of words satisfying the inductive properties described in A108081.
-/
inductive XWord : Word → Prop
  | base : XWord [0]
  | step_left {u v : Word} (hu : XWord u) (hv : XWord v) : XWord (l u ++ v)
  | step_right {u v : Word} (hu : XWord u) (hv : XWord v) : XWord (u ++ r v)

/-- `xN n` is the set of words in `XWord` of length `n`. -/
def xN (n : ℕ) : Set Word :=
  {w : Word | XWord w ∧ w.length = n}

/--
"The number of words of length $n$ for $n \le 12$ is given by $a(n+1)$. Is this always true?"

Formalized as $|X_n| = a(n-1)$ for $n \ge 1$, because the sequence values $a(0)=1, a(1)=2, a(2)=7$
match the examples given for word lengths $n=1, 2, 3$ respectively.
-/
@[category research open, AMS 5]
theorem count_words_in_x_is_a_shifted (n : ℕ) :
    n ≥ 1 → Set.ncard (xN n) = a (n - 1) := by
  sorry

end OeisA108081

• Formalization source ↗

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A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.