The tribonacci sequence is a generalization of the Fibonacci numbers. It is defined by , , , , and the recurrence equation

(1)

for integers (e.g., Develin 2000), and represents the case of the Fibonacci n-step numbers. The terms of the tribonacci sequence are known as tribonacci numbers.

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, whose term at index is ).

With for , the partial sum of the first terms is

(2)

for (V. Ivlev, pers. comm., Jul. 14, 2007). This is the case of the partial sum identity for Fibonacci n-step numbers.

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). Noe and Post (2005, pp. 4-5 and 7) report primality proofs for the nine terms whose indices are at most 10000.

Using Brown's criterion, it can be shown that the tribonacci numbers are complete; that is, every positive integer can be written as the sum of distinct tribonacci numbers. More precisely, every positive integer has a unique expansion as a sum of terms with distinct indices , with no three consecutive indices used (Carlitz et al. 1972). This analog of a Zeckendorf representation can be computed using a greedy algorithm.

An exact expression for , for nonnegative integers , is given by

			(3)
			(4)

where are the three roots of the polynomial

(5)

This can be written in slightly more concise form as

(6)

where , , and (Komatsu and Li 2019, p. 29). These coefficients are the three roots of the polynomial

(7)

The generating function is

(8)

for , where is the tribonacci constant.

Another explicit formula for is also given by

(9)

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 can use the LLL algorithm.

The ratio tends to the positive real root , namely 1.83928... (OEIS A058265), sometimes known as the tribonacci constant.

By considering the series (mod ), one can prove that any positive integer is a factor of for some positive integer (Brenner 1954). The smallest values of for which is a factor for , 2, ... are given by 1, 3, 7, 4, 14, 7, 5, 7, 9, 19, 8, 7, 6, ... (OEIS A112305).

The tribonacci constant is extremely prominent in the properties of the snub cube, its dual the pentagonal icositetrahedron, and the snub cube-pentagonal icositetrahedron compound. It can even be used to express the hard hexagon entropy constant.

With different initial values, the tribonacci sequence starts as , , , , , , , , ..., which gives the following sequences as special cases.

			OEIS	sequence
0	0	1	A000073	0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ...
1	1	1	A000213	1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, ...
0	1	0	A001590	0, 1, 0, 1, 2, 3, 6, 11, 20, 37, 68, 125, 230, ...
3	1	3	A001644	3, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, ...
	2	2	A100683	, 2, 2, 3, 7, 12, 22, 41, 75, 138, 254, 467, ...