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.
Dawsey, Rajat Gupta, Noah Lebowitz-Lockard, and Joseph Vandehey
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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.