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

formal-conjectures · oeis · ams-5

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.

"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.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A108081
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/108081.lean
Local target: Corpus.OEIS108081.count_words_in_x_is_a_shifted
Source SHA-256: 859dce3e11c79ab3af091c812e3a060ee9e29af438c11d10a79767bbe7e26264
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record