181546.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

10 declarations, 1 contain sorry, 0 declared as axioms

Sum of fourth powers of Fibonacci-like binomial coefficients

The sequence is defined by $$a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4.$$

References:

A181546

Imports (1)

FormalConjecturesUtil

def OeisA181546.F The generalized sum $F(n, L) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^L$ line 36 source
Source documentation
The generalized sum $F(n, L) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^L$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat, HPow.hPow, Nat.choose, HSub.hSub
12 instances, coercions and auxiliary constants left out.
```
def F (n L : ℕ) : ℕ :=
…
```
def OeisA181546.a $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4$ line 40 source
Source documentation
$a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA181546.F, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ := F n 4
```
Mentions: OeisA181546.F
def OeisA181546.lucas Lucas numbers $L(0) = 2, L(1) = 1, L(n) = L(n-1) + L(n-2)$ line 43 source
Source documentation
Lucas numbers $L(0) = 2, L(1) = 1, L(n) = L(n-1) + L(n-2)$.

no axioms what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def lucas : ℕ → ℕ
  | 0 => 2
  | 1 => 1
  | n + 2 => lucas (n + 1) + lucas n

/-- Conjectured limit value for $F(n+1, L) / F(n, L)$. -/
```
Mentions: OeisA181546.F
def OeisA181546.limitValue Conjectured limit value for $F(n+1, L) / F(n, L)$ line 49 source
Source documentation
Conjectured limit value for $F(n+1, L) / F(n, L)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HDiv.hDiv, HAdd.hAdd, HMul.hMul, Nat.fib, Real.sqrt, OfNat.ofNat, OeisA181546.lucas
12 instances, coercions and auxiliary constants left out.
```
noncomputable def limitValue (L : ℕ) : ℝ :=
…
```
theorem OeisA181546.a_0 Value of the sequence a at 0 line 54 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA181546.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by decide
…
```
Mentions: OeisA181546.a
theorem OeisA181546.a_1 Value of the sequence a at 1 line 58 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA181546.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by decide
…
```
Mentions: OeisA181546.a
theorem OeisA181546.a_2 Value of the sequence a at 2 line 62 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA181546.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 2 := by decide
…
```
Mentions: OeisA181546.a
theorem OeisA181546.a_3 Value of the sequence a at 3 line 66 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA181546.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 17 := by decide
…
```
Mentions: OeisA181546.a
theorem OeisA181546.a_4 Value of the sequence a at 4 line 70 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA181546.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 83 := by decide
…
```
Mentions: OeisA181546.a
theorem OeisA181546.conjecture sorry line 79 source
depends on sorry what this means
Statement uses: Nat, Filter.Tendsto, Real, HDiv.hDiv, OeisA181546.F, HAdd.hAdd, OfNat.ofNat, Filter.atTop, nhds, OeisA181546.limitValue
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

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