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The first few terms using the above indexing convention for , 1, 2, ... are 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ... (OEIS A000073; which however adopts the alternate indexing convention and ).

The first few prime tribonacci numbers are 2, 7, 13, 149, 19341322569415713958901, ... (OEIS A092836), which have indices 3, 5, 6, 10, 86, 97, 214, 801, 4201, 18698, 96878, ... (OEIS A092835), and no others with (E. W. Weisstein, Mar. 21, 2009).

Using Brown's criterion, it can be shown that the tribonacci numbers are complete; that is, every positive number can be written as the sum of distinct tribonacci numbers. Moreover, every positive number has a unique Zeckendorf-like expansion as the sum of distinct tribonacci numbers and that sum does not contain three consecutive tribonacci numbers. The Zeckendorf-like expansion can be computed using a greedy algorithm.

where denotes the nearest integer function (Plouffe). The first part of the numerator is related to the real root of , but determination of the denominator requires an application of the LLL algorithm.

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