KNOTT-6.3 — Binet, Lucas, Catalan, d’Ocagne, Honsberger & Fibonomial Power Identities

Binet’s closed forms Fₙ = (Φⁿ − (−φ)ⁿ)/√5 and Lₙ = Φⁿ + (−φ)ⁿ place (Lₙ, Fₙ) on the alternating Pell-Lucas hyperbolas Lₙ² − 5 Fₙ² = 4(−1)ⁿ with unit sector triangles, unify Catalan, d’Ocagne, and Honsberger identities, and annihilate k-th powers Fₙᵏ via signed Fibonomial coefficients.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-6.3
Curated Source	Dr. Ron Knott, Fibonacci and Phi Formulae (#defs)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.identities.construct_lucas_binet_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/divine, geometor/wildberger

Constructive Synthesis & Exact Proof

On fibFormulae.html (KnottFormulae), Dr. Ron Knott organizes the algebraic web linking Jacques Binet’s closed forms, Édouard Lucas’s companion sequence \(L_n = \Phi^n + (-\phi)^n\), Eugène Catalan’s, d’Ocagne’s, and Ross Honsberger’s (Honsberger1985) addition laws, and signed Fibonomial power recurrences (Vajda1989).

• Pell–Lucas Alternating Hyperbola & Asymptote: Plotting \(P_n = (L_n, F_n)\) from origin O at P0 \((2, 0)\), P1 \((1, 1)\), P2 \((3, 1)\), P3 \((4, 2)\), and P4 \((7, 3)\) places every even/odd point on the conjugate hyperbolas \(L_n^2 - 5 F_n^2 = 4(-1)^n\) approaching golden asymptote OA of slope \(1/\sqrt{5}\).

• Unit-Area Sector Invariant & Fibonomials: Because \(L_n F_{n+1} - F_n L_{n+1} = 2(-1)^n\), consecutive sector triangles OP0P1, OP1P2, OP2P3, and OP3P4 bounded by chords P0P1 – P3P4 are equal in unsigned area (exact area \(1\)), while Fibonomial coefficients \(\binom{k+1}{j}_F\) annihilate \(k\)-th powers \(F_n^k\) identically.

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
fibonomial 5 2	\(15\)
fibonomial 6 3	\(60\)
F10	\(55\)
L10	\(123\)
F20 via F10 L10	\(6765\)
pell lucas sector area	\(1\)
asymptote slope	\(\frac{\sqrt{5}}{5}\)