Alphabeta MathExamples: Linear Recurrences and Rational Generating Functions
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 15 results · all verified · 9 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
Combinatorics
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Example
The Fibonacci generating function and Binet formula over ℚ(√5)
Example
The Lucas generating function and its two-root closed form
Example
The repeated pole (1-2x)⁻² produces the sequence (n+1)2ⁿ
Example
The Fibonacci companion matrix advances (Fₙ,Fₙ₊₁)
Example
Monomer–domino tilings of a board give the Fibonacci recurrence and transfer matrix
Example
Cyclic monomer–domino tilings are counted by Lucas numbers through a matrix trace
Example
Closed walks of length n in Kₚ are counted by (p-1)ⁿ+(p-1)(-1)ⁿ
Example
Binary words avoiding 11 are counted by Fₙ₊₂
Example
Binary words avoiding 101 have generating function (1+x²)/(1-2x+x²-x³)
Example
North–east–west walks without immediate horizontal reversal satisfy aₙ=2aₙ₋₁+aₙ₋₂
Counterexample
A recurrence over ℚ can require a proper splitting field for its exponential closed form
False statement
FALSE: The recurrence coefficients alone determine the sequence
False statement
FALSE: A split characteristic polynomial always gives a linear combination of pure exponentials
Counterexample
∑_n≥0n!xⁿ is a formal power series that is not rational over ℚ
False statement
FALSE: The transfer-matrix identity requires a spectral-radius or convergence hypothesis
Sources
- R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Example 4.1.2
- R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Example 4.7.16
- B. E. Sagan, Combinatorics: The Art of Counting, Section 3.7
- M. Waldschmidt, Linear Recurrence Sequences VI, slides 16-18
- R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Example 4.7.5
- R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Proposition 4.7.8
- R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Example 4.1.3
- R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Sections 4.1-4.2
- R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Theorem 4.7.2