Wu’s Pyramidal Complement Formula

Abstract

Wu’s formula enumerates every positive integer outside a k-gonal-pyramidal sequence for k at least nine.

Definition 1.1 (The k-gonal-pyramidal sequence).

$$\forall k\in \mathbb{N},\forall m\in \mathbb{N},P\left(k,m\right)=\left(k-2\right)\cdot \mathrm{choose}\left(m+1,3\right)+\mathrm{choose}\left(m+1,2\right)$$

Formalization. D5/S3/Arith/WuPyramidalComplement.pyramidal (✓ std3).

Citation. Chai Wah Wu (2025). Algorithms for Complementary Sequences. DOI: 10.5281/zenodo.17535229. URL: https://math.colgate.edu/~integers/z95/z95.pdf.

Commentary.

The binomial expression is integral at every natural index and equals m(m+1)(m(k-2)-(k-5))/6. The value at index zero is included only as a counting extension; the source sequence uses positive indices.

Definition 1.2 (The positive complement).

$$\forall k\in \mathbb{N},\forall x\in \mathbb{N},C\left(k,x\right)\iff \left(0<x\wedge \left(\forall m\in \mathbb{N},0<m\Rightarrow P\left(k,m\right)\ne x\right)\right)$$

Formalization. D5/S3/Arith/WuPyramidalComplement.complement (✓ std3).

Citation. Chai Wah Wu (2025). Algorithms for Complementary Sequences. DOI: 10.5281/zenodo.17535229. URL: https://math.colgate.edu/~integers/z95/z95.pdf.

Commentary.

C(k,x) holds exactly when x is positive and is not P(k,m) for any positive m. This definition is independent of the proposed enumeration formula.

Definition 1.3 (The upper threshold).

$$\forall k\in \mathbb{N},\forall h\in \mathbb{N},U\left(k,h\right)=\left(k-2\right)\cdot h^3+3\cdot \left(k-1\right)\cdot h^2+\left(2\cdot k-1\right)\cdot h+6$$

Formalization. D5/S3/Arith/WuPyramidalComplement.upperThreshold (✓ std3).

Citation. Chai Wah Wu (2025). Algorithms for Complementary Sequences. DOI: 10.5281/zenodo.17535229. URL: https://math.colgate.edu/~integers/z95/z95.pdf.

Commentary.

U(k,h) is the inclusive upper-branch threshold in Equation (6).

Definition 1.4 (The lower threshold).

$$\forall k\in \mathbb{N},\forall h\in \mathbb{N},L\left(k,h\right)=h\cdot \left(h-1\right)\cdot \left(h\cdot \left(k-2\right)+k+1\right)$$

Formalization. D5/S3/Arith/WuPyramidalComplement.lowerThreshold (✓ std3).

Citation. Chai Wah Wu (2025). Algorithms for Complementary Sequences. DOI: 10.5281/zenodo.17535229. URL: https://math.colgate.edu/~integers/z95/z95.pdf.

Commentary.

L(k,h) is the inclusive lower-branch threshold in Equation (6).

Definition 1.5 (The ordered source branches).

$$B\left(u,l,c,k,n,h\right)=\mathrm{if}\left(U\left(k,h\right)\le 6\cdot n,u,\mathrm{if}\left(6\cdot n\le L\left(k,h\right),l,c\right)\right)$$

Formalization. D5/S3/Arith/WuPyramidalComplement.branchSelector (✓ std3).

Citation. Chai Wah Wu (2025). Algorithms for Complementary Sequences. DOI: 10.5281/zenodo.17535229. URL: https://math.colgate.edu/~integers/z95/z95.pdf.

Commentary.

The upper test is evaluated first. If it fails, the lower test is evaluated; otherwise the middle code is returned.

Definition 1.6 (The three branch adjustments).

$$E\left(b,n,h\right)=\mathrm{match}\left(b,n+h+1,n+h-1,n+h\right)$$

Formalization. D5/S3/Arith/WuPyramidalComplement.evaluateBranch (✓ std3).

Citation. Chai Wah Wu (2025). Algorithms for Complementary Sequences. DOI: 10.5281/zenodo.17535229. URL: https://math.colgate.edu/~integers/z95/z95.pdf.

Commentary.

The codes some(true), some(false), and none evaluate respectively to n+h+1, n+h-1 using natural subtraction, and n+h.

Theorem 1.7 (Conjecture 1).

$$\forall k\in \mathbb{N},\forall n\in \mathbb{N},9\le k\Rightarrow \left(1\le n\Rightarrow \mathrm{nth}\left(C\left(k\right),n-1\right)=E\left(B\left(\mathrm{some}\left(true\right),\mathrm{some}\left(false\right),none,k,n,\lfloor {\frac{6\cdot n}{k-2}}^{\frac{1}{3}}\rfloor \right),n,\lfloor {\frac{6\cdot n}{k-2}}^{\frac{1}{3}}\rfloor \right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Arith/WuPyramidalComplement.wu_conjecture_one (✓ std3). ∎

Resolves. Problems/wu-pyramidal-complement-conjecture-one (proved) by D5/S3/Arith/WuPyramidalComplement.wu_conjecture_one.

Source. Repository-derived.

Acknowledgement. Chai Wah Wu (2025). Algorithms for Complementary Sequences. DOI: 10.5281/zenodo.17535229. URL: https://math.colgate.edu/~integers/z95/z95.pdf.

Commentary.

Let h be the floor of the real cube root of 6n/(k-2). For every k at least nine and every positive n, the (n-1)-st zero-based member of the positive complement is the value selected by the exact inclusive thresholds. The proof identifies this real floor with the corresponding integer cube-root index, locates the answer strictly between consecutive pyramidal values in all three branches, and counts exactly n-1 complement values below it.

References

• Truth anchor: D5/S3/Arith/WuPyramidalComplement.branchSelector
• Truth anchor: D5/S3/Arith/WuPyramidalComplement.complement
• Truth anchor: D5/S3/Arith/WuPyramidalComplement.evaluateBranch
• Truth anchor: D5/S3/Arith/WuPyramidalComplement.lowerThreshold
• Truth anchor: D5/S3/Arith/WuPyramidalComplement.pyramidal
• Truth anchor: D5/S3/Arith/WuPyramidalComplement.upperThreshold
• Truth anchor: D5/S3/Arith/WuPyramidalComplement.wu_conjecture_one
• Dependency: D5/S3/ConceptDynamics/InformationEscape/RegistrationTemplates
• Dependency: D5/S3/ConceptDynamics/RegistrationWitnesses