227582.lean`

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9 declarations, 1 contain sorry, 0 declared as axioms

Representation of sequence terms by harmonic numbers $\lfloor \frac{1}{2H(n) - H(n^2+n-1) - \gamma} \rfloor$

A227582: Expansion of $(2+3x+2x^2+2x^3+3x^4+x^5-x^6)/(1-2x+x^2-x^5+2x^6-x^7)$.

The sequence satisfies the linear recurrence: $$a(n) = 2 a(n-1) - a(n-2) + a(n-5) - 2 a(n-6) + a(n-7)$$ with initial values $a(1) = 2, a(2) = 7, a(3) = 14, a(4) = 23, a(5) = 35, a(6) = 50, a(7) = 67$.

The sequence is 1-indexed in OEIS, so $a(n)$ is the $(n-1)$-th term of the 0-indexed solution.

References:

A227582
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA227582.baseSeq The sequence $b_n$ such that $A227582(n) = b_{n-1}$ for $n \ge 1$ line 47 source
Source documentation
The sequence $b_n$ such that $A227582(n) = b_{n-1}$ for $n \ge 1$. This is the 0-indexed solution to the linear recurrence in $\mathbb{Z}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: OfNat.ofNat, Fin, Matrix.vecCons, Neg.neg, Matrix.vecEmpty, LinearRecurrence, LinearRecurrence.mk, LinearRecurrence.mkSol
4 instances, coercions and auxiliary constants left out.
```
def baseSeq (n : ℕ) : ℤ :=
…
```
def OeisA227582.a Expansion of $(2+3x+2x^2+2x^3+3x^4+x^5-x^6)/(1-2x+x^2-x^5+2x^6-x^7)$, 1-indexed line 60 source
Source documentation
Expansion of $(2+3x+2x^2+2x^3+3x^4+x^5-x^6)/(1-2x+x^2-x^5+2x^6-x^7)$, 1-indexed.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, LT.lt, OfNat.ofNat, Int.toNat, OeisA227582.baseSeq, HSub.hSub
5 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
lemma OeisA227582.a_1 line 68 source
extra axioms: _private.0.OeisA227582.a_1._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA227582.a, OfNat.ofNat
Proof uses: id, of_decide_eq_true, Int.toNat, OeisA227582.baseSeq
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA227582.a_2 line 71 source
extra axioms: _private.0.OeisA227582.a_2._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA227582.a, OfNat.ofNat
Proof uses: id, of_decide_eq_true, Int.toNat, OeisA227582.baseSeq
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA227582.a_3 line 74 source
extra axioms: _private.0.OeisA227582.a_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA227582.a, OfNat.ofNat
Proof uses: id, of_decide_eq_true, Int.toNat, OeisA227582.baseSeq
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA227582.a_4 line 77 source
extra axioms: _private.0.OeisA227582.a_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA227582.a, OfNat.ofNat
Proof uses: id, of_decide_eq_true, Int.toNat, OeisA227582.baseSeq
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA227582.a_5 line 80 source
extra axioms: _private.0.OeisA227582.a_5._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA227582.a, OfNat.ofNat
Proof uses: id, of_decide_eq_true, Int.toNat, OeisA227582.baseSeq
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA227582.baseSeq_recurrence The linear recurrence: $$b(n + 7) = 2 b(n + 6) - b(n + 5) + b(n + 2) - 2 b(n + 1) + b(n)$$ holds for baseSeq line 88 source
Source documentation
The linear recurrence: $$b(n + 7) = 2 b(n + 6) - b(n + 5) + b(n + 2) - 2 b(n + 1) + b(n)$$ holds for baseSeq. This follows directly from LinearRecurrence.is_sol_mkSol.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Int, OeisA227582.baseSeq, HAdd.hAdd, OfNat.ofNat, HSub.hSub, HMul.hMul
Proof uses: LinearRecurrence, LinearRecurrence.mk, Matrix.vecCons, Neg.neg, Matrix.vecEmpty, Fin, id, LinearRecurrence.mkSol, LinearRecurrence.order, Finset.sum, Finset.univ, LinearRecurrence.coeffs, LinearRecurrence.is_sol_mkSol, Eq.mpr, congrArg, congrFun', Eq.trans, Finset.sum_congr, Eq.refl, Finset and 47 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem baseSeq_recurrence (n : ℕ) :
    baseSeq (n + 7) = 2 * baseSeq (n + 6) - baseSeq (n + 5) +
    baseSeq (n + 2) - 2 * baseSeq (n + 1) + baseSeq n := by
…
```
Mentions: OeisA227582.baseSeq
theorem OeisA227582.a_eq_floor_harmonic_expr sorry line 122 source
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Eq, OeisA227582.a, Int.toNat, Int.floor, Real, HDiv.hDiv, HSub.hSub, HMul.hMul, harmonic, HAdd.hAdd, Real.eulerMascheroniConstant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.