The Lucas Number Sequence

The Lucas number sequence, a cousin of the more famous Fibonacci sequence, is defined by the starting values L(0) = 2 and L(1) = 1, followed by the recurrence relation L(n) = L(n-1) + L(n-2). This deceptively simple rule generates a number sequence whose mathematical properties, especially its convergence toward the golden ratio (φ), create a natural framework for analyzing growth patterns and valuation multiples.

For stock analysis, it is not the sequence itself that is of interest, but the underlying principles of exponential growth and convergence that it represents.

The most direct application is to use the quotient between two consecutive numbers in the sequence (L(n)/L(n-1)) as a frame of reference for valuation multiples such as P/E, P/S or EV/EBITDA. For a young, fast-growing tech company, a P/E ratio approaching φ (1.618) from a previous, much higher level (e.g.

2.618, 4.236) can be interpreted as an indicator that the company has passed its most aggressive expansion phase and is approaching a more mature, more stable growth. This can signal a time point for a reinterpretation of the valuation, from a growth premium to a valuation based on future cash flows.

A more advanced application is to use the Lucas sequence to create a 'growth path' for a company's revenue or users. By adapting the recursive formula L(n) = L(n-1) + L(n-2) to company-specific data, one can create a model that projects a theoretical, 'natural' growth curve. By comparing the actual result with this mathematical curve, analysts can identify deviations.

A significant downward deviation can indicate a temporary problem or a fundamental change in the business model, while an upward deviation can signal a unique market position or a disruptive innovation not captured by traditional models.