258667.lean`

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

12 declarations, 1 contain sorry, 0 declared as axioms

Asymptotics of an inner sum formula for $a(n)$

The inner sum of the formula used in a: $$\sum_{\max(k-n+5, 0) \le j \le \min(k,4)} \binom{8-j}{j}\binom{2n-k+j-10}{k-j}$$

a: A total of $n$ married couples, including a mathematician M and his wife, are to be seated at the $2n$ chairs around a circular table, with no man seated next to his wife. After the ladies are seated at every other chair, M is the first man allowed to choose one of the remaining chairs. The sequence gives the number of ways of seating the other men, with no man seated next to his wife, if M chooses the chair that is 9 seats clockwise from his wife's chair.

$$a(n) = \begin{cases} 0 & \text{if } n \le 5 \cr \sum_{k=0}^{n-1}(-1)^k(n-k-1)! \sum_{\max(k-n+5, 0) \le j \le \min(k,4)} \binom{8-j}{j}\binom{2n-k+j-10}{k-j} & \text{if } n > 5 \end{cases}$$

References:

A258667
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA258667.inner_sum The inner sum of the formula used in a: $$\sum_{\max(k-n+5, 0) \le j \le \min(k,4)} \binom{8-j}{j}\binom{2n-k+j-10}{k-j}$$ line 54 source
Source documentation
The inner sum of the formula used in a: $$\sum_{\max(k-n+5, 0) \le j \le \min(k,4)} \binom{8-j}{j}\binom{2n-k+j-10}{k-j}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Max.max, OfNat.ofNat, HSub.hSub, HAdd.hAdd, Min.min, Finset.sum, Finset.Icc, Nat.choose, HMul.hMul, Int.ofNat
13 instances, coercions and auxiliary constants left out.
```
def inner_sum (n k : ℕ) : ℤ :=
…
```
def OeisA258667.a a: A total of $n$ married couples, including a mathematician M and his wife, are to be seated at the $2n$ chairs around a circular table, with no man seated next to his wife line 74 source
Source documentation
a: A total of $n$ married couples, including a mathematician M and his wife, are to be seated at the $2n$ chairs around a circular table, with no man seated next to his wife. After the ladies are seated at every other chair, M is the first man allowed to choose one of the remaining chairs. The sequence gives the number of ways of seating the other men, with no man seated next to his wife, if M chooses the chair that is 9 seats clockwise from his wife's chair.
$$a(n) = \begin{cases} 0 & \text{if } n \le 5 \cr \sum_{k=0}^{n-1}(-1)^k(n-k-1)! \sum_{\max(k-n+5, 0) \le j \le \min(k,4)} \binom{8-j}{j}\binom{2n-k+j-10}{k-j} & \text{if } n > 5 \end{cases}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, LE.le, OfNat.ofNat, Int.natAbs, Finset.sum, Int, Finset.range, Eq, HMod.hMod, Neg.neg, Int.ofNat, Nat.factorial, HSub.hSub, HMul.hMul, OeisA258667.inner_sum
13 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA258667.natFacToReal line 84 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Nat.factorial
2 instances, coercions and auxiliary constants left out.
def OeisA258667.menageDenomTerm The denominator term $k! (n-1)_k$ represented as a Real number line 87 source
Source documentation
The denominator term $k! (n-1)_k$ represented as a Real number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: OeisA258667.natFacToReal, Nat.descFactorial, HSub.hSub, OfNat.ofNat, HMul.hMul
7 instances, coercions and auxiliary constants left out.
```
noncomputable def menageDenomTerm (n k : ℕ) : ℝ :=
…
```
def OeisA258667.asymptoticSumPart The infinite series part of the asymptotic expansion: $\sum_{k \ge 1} \frac{(-1)^k}{k!(n-1)_k}$ line 95 source
Source documentation
The infinite series part of the asymptotic expansion: $\sum_{k \ge 1} \frac{(-1)^k}{k!(n-1)_k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Finset.sum, Finset.range, ite, Eq, OfNat.ofNat, OeisA258667.menageDenomTerm, HDiv.hDiv, HPow.hPow, Neg.neg
16 instances, coercions and auxiliary constants left out.
```
noncomputable def asymptoticSumPart (n : ℕ) : ℝ :=
…
```
def OeisA258667.asymptoticTerm The proposed asymptotic expression for a(n) line 106 source
Source documentation
The proposed asymptotic expression for a(n).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: ite, LE.le, OfNat.ofNat, OeisA258667.natFacToReal, HMul.hMul, Real.exp, Neg.neg, HDiv.hDiv, HSub.hSub, HAdd.hAdd, OeisA258667.asymptoticSumPart
21 instances, coercions and auxiliary constants left out.
```
noncomputable def asymptoticTerm (n : ℕ) : ℝ :=
…
```
lemma OeisA258667.a_1 line 116 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA258667.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA258667.a_2 line 119 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA258667.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA258667.a_3 line 122 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA258667.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA258667.a_4 line 125 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA258667.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA258667.a_5 line 128 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA258667.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA258667.a_is_equivalent_asymptoticTerm sorry line 139 source
depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, OeisA258667.a, OeisA258667.asymptoticTerm
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

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