The Padovan sequence: ratios converge to rho
Padovan: 1,1,1,2,2,3,4,5,7,9,12,16,21... each term = sum two and three steps back. Ratios converge to rho.
Golden ratio and plastic number: the self-similarity equations
φ² = φ + 1 → φ ≈ 1.61803 (2-D)
ρ³ = ρ + 1 → ρ ≈ 1.32471 (3-D)
Both are the unique real roots > 1 of their respective equations. ρ generalises φ from 2-term to 3-term recurrences.
Padovan sequence: each bar is sum of the two bars before the previous
The Padovan sequence 1,1,1,2,2,3,4,5,7,9,12… each term = term two ago + term three ago. The bars grow asymptotically at rate ρ ≈ 1.3247 per step. The golden ratio governs 2-step Fibonacci; the plastic number governs this 3-step variant.
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