303639.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 two squares and two central-binomial-type terms

A303639 counts the ways to write $n$ as $a^2 + b^2 + \binom{2c+1}{c} + \binom{2d+1}{d}$ with $a, b, c, d$ nonnegative integers, $a \le b$ and $c \le d$.

References:

A303639
Z.-W. Sun, "Restricted sums of four squares", arXiv:1701.05868 [math.NT], 2017-2018.
Z.-W. Sun, "Refining Lagrange's four-square theorem", J. Number Theory 175 (2017), 167-190.
S. Jeong, A303639 Counterexample Project, search, certificates and independent verifiers, DOI 10.5281/zenodo.21863025. The counterexample is recorded as an approved comment on the OEIS entry (Aug 10 2026).

Imports (1)

FormalConjecturesUtil

def OeisA303639.A The predicate that $n$ can be written as $a^2 + b^2 + \binom{2c+1}{c} + \binom{2d+1}{d}$ for nonnegative integers $a, b, c, d$ line 48 source
Source documentation
The predicate that $n$ can be written as $a^2 + b^2 + \binom{2c+1}{c} + \binom{2d+1}{d}$ for nonnegative integers $a, b, c, d$.
This drops the normalisations $a \le b$ and $c \le d$. They do not affect whether the count is positive, and refuting the unordered statement is the stronger result, since a representation with $a \le b$ and $c \le d$ is in particular a representation.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, Nat.choose, HMul.hMul
9 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA303639.a_1 line 52 source
standard axioms: Quot.sound, propext what this means
Statement uses: Not, OeisA303639.A, OfNat.ofNat, Nat
Proof uses: Exists, Eq, HAdd.hAdd, HPow.hPow, Nat.choose, HMul.hMul, False, LT.lt, Nat.choose_pos, Decidable.byContradiction, LE.le
17 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA303639.a_2 line 59 source
standard axioms: propext what this means
Statement uses: OeisA303639.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Nat.choose, HMul.hMul, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303639.a_3 line 63 source
standard axioms: propext what this means
Statement uses: OeisA303639.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Nat.choose, HMul.hMul, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303639.a_4 line 67 source
standard axioms: propext what this means
Statement uses: OeisA303639.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Nat.choose, HMul.hMul, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303639.a_5 line 71 source
standard axioms: propext what this means
Statement uses: OeisA303639.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Nat.choose, HMul.hMul, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303639.a_6 line 75 source
standard axioms: propext what this means
Statement uses: OeisA303639.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Nat.choose, HMul.hMul, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303639.conjecture sorry line 95 source
depends on sorry what this means
Statement uses: Not, Nat, LT.lt, OfNat.ofNat, OeisA303639.A
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303639.conjecture.counterexample sorry line 101 source
depends on sorry what this means
Statement uses: Not, OeisA303639.A, OfNat.ofNat, Nat
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.