We consider s = X³ + X + 1, so n = 4.
First, we compute f and t.

f = g₀ + _j=1³ (g_j - _i=1^j   s_4-i · g_j-i)X^j.
= g₀ + (g₁ - s₃g₀)X + (g₂ - s₃g₁ - s₂g₀)X² + (g₃ - s₃g₂ - s₂g₁ - s₁g₀)X³
= g₀ + (g₁ - g₀)X + (g₂ - g₁)X² + (g₃ - g₂ - g₀)X³

and

t = s₃X + s₂X² + s₁X³ + s₀X⁴ = X + X³ + X⁴.

Working out the formula for g_i with i < 15, and discarding the terms with exponent >14, we find

(1 + t + t² + ··· + t^k) f = g₀ + g₁X + g₂X² + g₃X³ + (g₁ + g₃ + g₀)X ⁴ + (2g₁ + g₃ + g2 + g₀)X⁵ + (2g₁ + 2g₃ + 2g₂ + g₀)X⁶ + (3g₁ + 4g₃ + 2g₂ + 2g₀)X⁷ + (6g₁ + 6g₃ + 3 g₂ + 4g₀)X⁸ + (10g₁ + 9g₃ + 6 g₂ + 6g₀)X⁹ + (15g₁ + 15g₃ + 10g₂ + 9g₀)X¹⁰ + (24g₁ + 25g₃ + 15g₂ + 15g₀)X¹¹ + (40g₁ + 40g₃ + 24g₂ + 25g₀)X¹² + (65g₁ + 64g₃ + 40g₂ + 40g₀ )X¹³ + (104g₁ + 104g₃ + 65g₂ + 64g₀)X¹⁴ + ···

We read off from this polynomial that g₁₄ = 104g₁ + 104g₃ + 65g₂ + 64g₀.

The Fibonacci sequence g₀, g₁, g₂, g₃, .... is determined by the recurrence g_k+2 = g_k + g_k+1.

The sequence is only fully determined once we specify the two first numbers:

g₀ = a,     g₁ = b.

The sequence can be produced by means of the shift register with shift polynomial s = X + 1 Z[X].

According to the above formula, with f = a + (b - a)X and t = X + X², the entry g_i is given as the coefficient of Xⁱ in

(1 + t + t² + ··· ) (a + (b-a)X) = a + bX + (a + b)X² + (a + 2b)X³ + (2a+3b)X⁴ + (3a + 5b)X⁵ + ··· .

The alternative name rabbit sequence for the Fibonacci sequence comes from the interpretation of g_n as the number of rabbits of a given population in the year n. The recurrence g_k+2 = g_k+1 + g_k is inspired by their reproductive behaviour.