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import FormalConjecturesUtilSymmetry of digit sum differences $\operatorname{sign}(S_5(k) - S_1(k))$
a: $\mathrm{a}(n) = \sum_{k=0}^n \mathrm{sign}(\mathrm{A007953}(5k) - \mathrm{A007953}(k))$. $\mathrm{A007953}(n)$ is the digital sum of $n$ in base 10. The sequence is non-negative, so the sum over $\mathbb{Z}$ is converted to $\mathbb{N}$.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA289411open Nata: $\mathrm{a}(n) = \sum_{k=0}^n \mathrm{sign}(\mathrm{A007953}(5k) - \mathrm{A007953}(k))$. $\mathrm{A007953}(n)$ is the digital sum of $n$ in base 10. The sequence is non-negative, so the sum over $\mathbb{Z}$ is converted to $\mathbb{N}$.
def a (n : ℕ) : ℕ :=
let digital_sum_ten (m : ℕ) : ℕ := (Nat.digits 10 m).sum
(Finset.range (n + 1)).sum (fun k =>
Int.sign ((digital_sum_ten (5 * k) : ℤ) - (digital_sum_ten k : ℤ)))
|>.toNat@[category test, AMS 11]
lemma a_0 : a 0 = 0 := ⊢ a 0 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 0 := ⊢ a 2 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 1 := ⊢ a 3 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 0 := ⊢ a 4 = 0 All goals completed! 🐙Conjecture: For $k \ge 1$, let $m_k = 10^k / 2 - 1$. Then for $i = 0, \ldots, m_k$, we have $a(m_k - i) = a(m_k + i)$.
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/289411.wip.lean#L167"]
theorem a_symm (k : ℕ) (hk : 0 < k) :
let m_k : ℕ := (10 ^ k) / 2 - 1
∀ i : ℕ, i ≤ m_k → a (m_k - i) = a (m_k + i) := k:ℕhk:0 < k⊢ let m_k := 10 ^ k / 2 - 1;
∀ i ≤ m_k, OeisA289411.a (m_k - i) = OeisA289411.a (m_k + i)
All goals completed! 🐙end OeisA289411