108306.lean`

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12 declarations, 1 contain sorry, 0 declared as axioms

A108306: Expansion of $(3x+1)/(1-3x-3x^2)$

This sequence satisfies the linear recurrence relation $a(0)=1$, $a(1)=6$, and $a(n) = 3a(n-1) + 3a(n-2)$ for $n \ge 2$.

References:

A108306

Imports (1)

FormalConjecturesUtil

def OeisA108306.a The primary defining sequence a line 37 source
Source documentation
The primary defining sequence a. a n is the $n$-th term of the expansion of $(3x+1)/(1-3x-3x^2)$, satisfying $a(0)=1, a(1)=6, a(n)=3a(n-1)+3a(n-2)$.

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

@[category test, AMS 11]
```
Mentions: test, AMS
theorem OeisA108306.a_0 line 43 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108306.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 OeisA108306.a_1 line 46 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108306.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 OeisA108306.a_2 line 49 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108306.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 OeisA108306.a_3 line 52 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108306.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 OeisA108306.a_4 line 55 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108306.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.
def OeisA108306.m The matrix for the specific case $a=5, b=2$ line 63 source
Source documentation
The matrix for the specific case $a=5, b=2$. $$M = \begin{pmatrix} 1 & 5 \cr 1 & 2 \end{pmatrix}$$

standard axioms: propext what this means
Statement uses: Matrix, Fin, OfNat.ofNat, Nat
Proof uses: OeisA108306.m.match_1, Unit
2 instances, coercions and auxiliary constants left out.
```
def m : Matrix (Fin 2) (Fin 2) ℕ :=
…
```
theorem OeisA108306.a_is_invert_transform_case The sequence is the INVERT transform of (1, 5, 10, 20, 40, 80, 160, ...) and can be obtained by extracting the upper left terms of matrix powers of [(1,5); (1,2)] line 76 source
Source documentation
The sequence is the INVERT transform of (1, 5, 10, 20, 40, 80, 160, ...) and can be obtained by extracting the upper left terms of matrix powers of [(1,5); (1,2)]. These results are a case (a=5, b=2) of the general conjecture below.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, OeisA108306.a, HPow.hPow, Matrix, Fin, OfNat.ofNat, OeisA108306.m, HAdd.hAdd
Proof uses: HSMul.hSMul, Eq.mpr, HMul.hMul, id, congrArg, pow_add, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, mul_add, mul_smul_comm, mul_one, Eq.symm, pow_succ, Nat.twoStepInduction, Nat.succ, OeisA108306.a.eq_3, Decidable.byContradiction and 3 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_is_invert_transform_case (n : ℕ) :
    a n = (m ^ (n + 1)) 0 0 := by
…
```
Mentions: OeisA108306.a, OeisA108306.m
def OeisA108306.invertSeqC The c sequence for the general INVERT transform conjecture line 93 source
Source documentation
The c sequence for the general INVERT transform conjecture. $c(1) = 1$, $c(k)=ab^(k-2)$ for $k \ge 2$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: OeisA108306.a.match_1, Unit, OfNat.ofNat, HMul.hMul, HPow.hPow
7 instances, coercions and auxiliary constants left out.
```
def invertSeqC (a b : ℕ) : ℕ → ℕ
  | 0 => 0
  | 1 => 1
  | (k + 2) => a * b^k

/-- The INVERT transform of the sequence c. -/
```
Mentions: OeisA108306.a
def OeisA108306.invertSeqD The INVERT transform of the sequence c line 99 source
Source documentation
The INVERT transform of the sequence c.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.strongRecOn, LT.lt, ite, Eq, OfNat.ofNat, Finset.sum, Finset.range, dite, HMul.hMul, OeisA108306.invertSeqC, HSub.hSub, Not
9 instances, coercions and auxiliary constants left out.
```
noncomputable def invertSeqD (a b : ℕ) (n : ℕ) : ℕ :=
…
```
Mentions: OeisA108306.a
def OeisA108306.genMatrix The general 2x2 matrix [(1,a); (1,b)] line 107 source
Source documentation
The general 2x2 matrix [(1,a); (1,b)].

standard axioms: propext what this means
Statement uses: Nat, Matrix, Fin, OfNat.ofNat
Proof uses: OeisA108306.m.match_1, Unit
2 instances, coercions and auxiliary constants left out.
```
def genMatrix (a b : ℕ) : Matrix (Fin 2) (Fin 2) ℕ :=
…
```
Mentions: OeisA108306.a
theorem OeisA108306.conjecture sorry line 122 source
depends on sorry what this means
Statement uses: Nat, Eq, OeisA108306.invertSeqD, HPow.hPow, Matrix, Fin, OfNat.ofNat, OeisA108306.genMatrix
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

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