107247.lean`

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

9 declarations, 2 contain sorry, 0 declared as axioms

Sum of squares of nonacci numbers

Sum of squares of nonacci numbers (Fibonacci 9-step numbers).

References:

A107247

Imports (1)

FormalConjecturesUtil

def OeisA107247.nonacci Nonacci number (Fibonacci 9-step number) line 36 source

Source documentation

Nonacci number (Fibonacci 9-step number). $F_9(n)$ is defined by the linear recurrence $F_9(n+9) = \sum_{i=0}^8 F_9(n+i)$, with $F_9(0) = \dots = F_9(7) = 0$ and $F_9(8) = 1$.

no axioms what this means

Statement uses: Nat

2 instances, coercions and auxiliary constants left out.

def nonacci : ℕ → ℕ
  | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0 | 4 => 0 | 5 => 0 | 6 => 0 | 7 => 0
  | 8 => 1
  | n + 9 =>
    nonacci (n + 8) + nonacci (n + 7) + nonacci (n + 6) + nonacci (n + 5) +
    nonacci (n + 4) + nonacci (n + 3) + nonacci (n + 2) + nonacci (n + 1) + nonacci n

/--
The primary defining sequence `a`.
`a n` is the sum of squares of nonacci numbers (Fibonacci 9-step numbers).
The official sequence matches the sum up to `n + 1` of the standard nonacci numbers
to align with the offset.

Mentions: OeisA107247.a

def OeisA107247.a The primary defining sequence a line 49 source
Source documentation
The primary defining sequence a. a n is the sum of squares of nonacci numbers (Fibonacci 9-step numbers). The official sequence matches the sum up to n + 1 of the standard nonacci numbers to align with the offset.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HPow.hPow, OeisA107247.nonacci
8 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA107247.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 54 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA107247.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 = 0 := by decide
…
```
Mentions: OeisA107247.a
theorem OeisA107247.a_1 line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA107247.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 OeisA107247.a_2 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA107247.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 OeisA107247.a_3 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA107247.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 OeisA107247.a_4 line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA107247.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 OeisA107247.known_prime_and_semiprimes Primes in this sequence include: $a(8) = 2$ sorry line 78 source
Source documentation
Primes in this sequence include: $a(8) = 2$. Semiprimes in this sequence include: $a(9) = 6 = 2 * 3$, $a(10) = 22 = 2 * 11$, $a(11) = 86 = 2 * 43$, $a(13) = 1366 = 2 * 683$, $a(14) = 5462 = 2 * 2731$, $a(16) = 87382 = 2 * 43691$, $a(17) = 348503 = 37 * 9419$, $a(27) = 358201316657 = 71 * 5045088967$. (Note: The OEIS comment uses 1-based indexing, so their indices are shifted by +1 compared to this formalization).

depends on sorry what this means
Statement uses: And, Nat.Prime, OeisA107247.a, OfNat.ofNat, Nat, Nat.IsSemiprime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem known_prime_and_semiprimes :
  (a 8).Prime ∧
  (a 9).IsSemiprime ∧
  (a 10).IsSemiprime ∧
  (a 11).IsSemiprime ∧
  (a 13).IsSemiprime ∧
  (a 14).IsSemiprime ∧
  (a 16).IsSemiprime ∧
  (a 17).IsSemiprime ∧
  (a 27).IsSemiprime := by
…
```
Mentions: OeisA107247.a
theorem OeisA107247.conjecture Primes in this sequence include: $a(8) = 2$, which is next? sorry line 94 source
Source documentation
Primes in this sequence include: $a(8) = 2$, which is next?

depends on sorry what this means
Statement uses: Eq, Nat, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, OeisA107247.a, InfSet.sInf, Set.ofPred, And, LT.lt, Nat.Prime
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
    answer(sorry) = a (sInf {n : ℕ | 8 < n ∧ (a n).Prime}) := by
…
```
Mentions: OeisA107247.a