100434.lean`

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L4DEL: Lean4 Domain Explanation Language

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

17 declarations, 3 contain sorry, 0 declared as axioms

Expansion of g.f. $(1+x)(3+x)/(1+6x^2+x^4)$

This sequence is defined by the linear recurrence relation $a(n) = -6 a(n-2) - a(n-4)$ for $n \ge 4$, with initial values $a(0)=3$, $a(1)=4$, $a(2)=-17$, $a(3)=-24$.

References:

A100434

Imports (1)

FormalConjecturesUtil

def OeisA100434.a The primary defining sequence a, which is the expansion of the generating function $(1+x)(3+x)/(1+6x^2+x^4)$ line 38 source
Source documentation
The primary defining sequence a, which is the expansion of the generating function $(1+x)(3+x)/(1+6x^2+x^4)$. It satisfies the recurrence $a(n) = -6 a(n-2) - a(n-4)$ for $n \ge 4$.

no axioms what this means
Statement uses: Nat, Int
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℤ
  | 0 => 3
  | 1 => 4
  | 2 => -17
  | 3 => -24
  | n + 4 => -6 * a (n + 2) - a n

/-- $c(n)$ starts with $(1, -3, -7, 17)$ and satisfies the same recurrence as `a` -/
```
Mentions: OeisA100434.c
def OeisA100434.c $c(n)$ starts with $(1, -3, -7, 17)$ and satisfies the same recurrence as a line 46 source
Source documentation
$c(n)$ starts with $(1, -3, -7, 17)$ and satisfies the same recurrence as a

no axioms what this means
Statement uses: Nat, Int
2 instances, coercions and auxiliary constants left out.
```
def c : ℕ → ℤ
  | 0 => 1
  | 1 => -3
  | 2 => -7
  | 3 => 17
  | n + 4 => -6 * c (n + 2) - c n

/-- $d(n)$ starts with $(2, 4, -10, -24)$ and satisfies the same recurrence as `a` -/
```
Mentions: OeisA100434.d, OeisA100434.a
def OeisA100434.d $d(n)$ starts with $(2, 4, -10, -24)$ and satisfies the same recurrence as a line 54 source
Source documentation
$d(n)$ starts with $(2, 4, -10, -24)$ and satisfies the same recurrence as a

no axioms what this means
Statement uses: Nat, Int
2 instances, coercions and auxiliary constants left out.
```
def d : ℕ → ℤ
  | 0 => 2
  | 1 => 4
  | 2 => -10
  | 3 => -24
  | n + 4 => -6 * d (n + 2) - d n

/-- $b(2n) = -c(2n+1)$, $b(2n+1) = c(2n)$ -/
-- Sign corrected (even branch): see https://github.com/google-deepmind/formal-conjectures/issues/5025
```
Mentions: OeisA100434.b, OeisA100434.c
def OeisA100434.b $b(2n) = -c(2n+1)$, $b(2n+1) = c(2n)$ line 63 source
Source documentation
$b(2n) = -c(2n+1)$, $b(2n+1) = c(2n)$

no axioms what this means
Statement uses: Nat, Int
Proof uses: ite, Eq, HMod.hMod, OfNat.ofNat, Neg.neg, OeisA100434.c, HAdd.hAdd, HSub.hSub
9 instances, coercions and auxiliary constants left out.
```
def b (n : ℕ) : ℤ :=
…
```
def OeisA100434.e $e(2n) = d(2n)/2$, $e(2n+1) = - d(2n)/2$ line 68 source
Source documentation
$e(2n) = d(2n)/2$, $e(2n+1) = - d(2n)/2$

no axioms what this means
Statement uses: Nat, Int
Proof uses: ite, Eq, HMod.hMod, OfNat.ofNat, HDiv.hDiv, OeisA100434.d, Neg.neg, HSub.hSub
10 instances, coercions and auxiliary constants left out.
```
def e (n : ℕ) : ℤ :=
…
```
def OeisA100434.f $f(2n) = f(2n+1) = d(2n+1)/2$ line 76 source
Source documentation
$f(2n) = f(2n+1) = d(2n+1)/2$

no axioms what this means
Statement uses: Nat, Int
Proof uses: HDiv.hDiv, OfNat.ofNat, OeisA100434.d, HAdd.hAdd, HMul.hMul
9 instances, coercions and auxiliary constants left out.
```
def f (n : ℕ) : ℤ :=
…
```
def OeisA100434.g $g(2n) = 0, g(2n+1) = c(2n+1)$ line 81 source
Source documentation
$g(2n) = 0, g(2n+1) = c(2n+1)$

no axioms what this means
Statement uses: Nat, Int
Proof uses: ite, Eq, HMod.hMod, OfNat.ofNat, OeisA100434.c
5 instances, coercions and auxiliary constants left out.
```
def g (n : ℕ) : ℤ :=
…
```
theorem OeisA100434.a_0 Value of the sequence a at 0 line 87 source
Source documentation
Value of the sequence a at 0.

no axioms what this means
Statement uses: Eq, Int, OeisA100434.a, OfNat.ofNat, Nat
Proof uses: 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 = 3 := by rfl
…
```
Mentions: OeisA100434.a
theorem OeisA100434.a_1 Value of the sequence a at 1 line 91 source
Source documentation
Value of the sequence a at 1.

no axioms what this means
Statement uses: Eq, Int, OeisA100434.a, OfNat.ofNat, Nat
Proof uses: 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 = 4 := by rfl
…
```
Mentions: OeisA100434.a
theorem OeisA100434.a_2 Value of the sequence a at 2 line 95 source
Source documentation
Value of the sequence a at 2.

no axioms what this means
Statement uses: Eq, Int, OeisA100434.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 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 = -17 := by rfl
…
```
Mentions: OeisA100434.a
theorem OeisA100434.a_3 Value of the sequence a at 3 line 99 source
Source documentation
Value of the sequence a at 3.

no axioms what this means
Statement uses: Eq, Int, OeisA100434.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 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 = -24 := by rfl
…
```
Mentions: OeisA100434.a
theorem OeisA100434.a_4 Value of the sequence a at 4 line 103 source
Source documentation
Value of the sequence a at 4.

no axioms what this means
Statement uses: Eq, Int, OeisA100434.a, OfNat.ofNat, Nat
Proof uses: 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 = 99 := by rfl
…
```
Mentions: OeisA100434.a
theorem OeisA100434.a_even For all $n \ge 0$, we have $a(2n) = - c(2n+1)$ line 109 source
Source documentation
For all $n \ge 0$, we have $a(2n) = - c(2n+1)$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Eq, Int, OeisA100434.a, HMul.hMul, OfNat.ofNat, Neg.neg, OeisA100434.c, HAdd.hAdd
Proof uses: Nat.twoStepInduction, Eq.refl, Decidable.byContradiction, Not, Eq.mpr, id, congrArg, HSub.hSub, Eq.symm, Mathlib.Tactic.Ring.of_eq, HPow.hPow, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne, inferInstance, Ring, Mathlib.Tactic.Ring.Common.sub_congr, Int.rawCast, Int.negOfNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.neg_congr and 25 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem a_even (n : ℕ) : a (2 * n) = - c (2 * n + 1) := by
…
```
Mentions: OeisA100434.a, OeisA100434.c
theorem OeisA100434.a_odd For all $n \ge 0$, we have $a(2n+1) = d(2n+1)$ line 134 source
Source documentation
For all $n \ge 0$, we have $a(2n+1) = d(2n+1)$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Eq, Int, OeisA100434.a, HAdd.hAdd, HMul.hMul, OfNat.ofNat, OeisA100434.d
Proof uses: Nat.twoStepInduction, Eq.refl, Decidable.byContradiction, Not, Eq.mpr, id, congrArg, HSub.hSub, Neg.neg, Eq.symm
18 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
```
theorem a_odd (n : ℕ) : a (2 * n + 1) = d (2 * n + 1) := by
…
```
Mentions: OeisA100434.a, OeisA100434.d
theorem OeisA100434.conjecture1 sorry line 165 source
depends on sorry what this means
Statement uses: Nat, Eq, Int, HAdd.hAdd, OeisA100434.c, OeisA100434.d, OeisA100434.b
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA100434.conjecture2 sorry line 181 source
depends on sorry what this means
Statement uses: Nat, Eq, Int, HAdd.hAdd, OeisA100434.e, OeisA100434.f, OeisA100434.b
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA100434.conjecture3 sorry line 197 source
depends on sorry what this means
Statement uses: Nat, Eq, Int, HAdd.hAdd, OeisA100434.g, OeisA100434.a, OeisA100434.b
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

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