B(0;k), the number of ways to write zero as a signed sum of the first k Fibonacci numbers, satisfies the Tribonacci recurrence B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2), with a modified version holding for general n.
Representations des nombres naturels par une somme de nombres de Fibonacci on de nombres de Lucas.Bulletin de La Society Royale des Sciences de Liege, pages 179–182
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Non-standard Zeckendorf decompositions; or, Tribonacci within Fibonacci
B(0;k), the number of ways to write zero as a signed sum of the first k Fibonacci numbers, satisfies the Tribonacci recurrence B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2), with a modified version holding for general n.