Abstract
Fibonacci and Lucas sequences are basic examples of second-order recurrences, and their behavior is closely
connected to the golden ratio. Bernoulli numbers and special values of the Riemann zeta function also form a
classical part of number theory. This paper connects these two areas through exact finite identities. The method starts
from exponential generating functions, separates the odd-indexed terms, applies Bernoulli generating functions, and
then compares coefficients. This gives a finite formula in which a weighted sum of zeta values at non-positive
integers becomes an explicit Fibonacci expression. The same argument also gives a Lucas version, and then extends
to every sequence satisfying the Fibonacci recurrence with arbitrary initial values. Exact symbolic checks and residual
plot are included to show how the cancellation works. The result is a complete unconditional link between Fibonaccitype
recurrences, Bernoulli numbers, and special zeta values.
