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Tribonacci Number


The tribonacci numbers are a generalization of the Fibonacci numbers defined by T_1=1, T_2=1, T_3=2, and the recurrence equation

 T_n=T_(n-1)+T_(n-2)+T_(n-3)
(1)

for n>=4 (e.g., Develin 2000). They represent the n=3 case of the Fibonacci n-step numbers.

The first few terms using the above indexing convention for n=0, 1, 2, ... are 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ... (OEIS A000073; which however adopts the alternate indexing convention T_0=T_1=0 and T_2=1).

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 n<=291217 (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.

An exact expression for the nth tribonacci number can be given explicitly by

T_n=(alpha^(n+1))/((alpha-beta)(alpha-gamma))+(beta^(n+1))/((beta-alpha)(beta-gamma))+(gamma^(n+1))/((gamma-alpha)(gamma-beta))
(2)
=(alpha^n)/(-alpha^2+4alpha-1)+(beta^n)/(-beta^2+4beta-1)+(gamma^n)/(-gamma^2+4gamma-1),
(3)

where (alpha,beta,gamma) are the three roots of the polynomial

 P(x)=x^3-x^2-x-1.
(4)

This can be written in slightly more concise form as

 T_n=r_1alpha^n+r_2beta^n+r_3gamma^n,
(5)

where r_n is the nth root of the polynomial

 Q(y)=44y^3-2y-1
(6)

and (alpha,beta,gamma) and (r_1,r_2,r_3) are in the ordering of the Wolfram Language's Root object.

The tribonacci numbers can also be computed using the generating function

 z/(1-z-z^2-z^3) 
=1+z+2z^2+4z^3+7z^4+13z^5+24z^6+44z^7+81z^8+149z^9+....
(7)

Another explicit formula for T_n is also given by

 [3({1/3(19+3sqrt(33))^(1/3)+1/3(19-3sqrt(33))^(1/3)+1/3}^n(586+102sqrt(33))^(1/3))/((586+102sqrt(33))^(2/3)+4-2(586+102sqrt(33))^(1/3))],
(8)

where [x] denotes the nearest integer function (Plouffe). The first part of the numerator is related to the real root of x^3-x^2-x-1, but determination of the denominator requires an application of the LLL algorithm.

The ratio of adjacent terms tends to the positive real root (x^3-x^2-x-1)_1, namely 1.83929... (OEIS A058265), sometimes known as the tribonacci constant.

By considering the series T_n (mod k), one can prove that any integer k is a factor of T_n for some n (Brenner 1954). The smallest values of n for which k is a factor for k=1, 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 a, b, c, a+b+c, a+2b+2c, 2a+3b+4c, 4a+6b+7c, 7a+11b+13c, ..., which gives the following sequences as special cases.

abcOEISsequence
001A0000730, 1, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ...
111A0002131, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, ...
010A0015900, 1, 0, 1, 2, 3, 6, 11, 20, 37, 68, 125, 230, ...
313A0016443, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, ...
-122A100683-1, 2, 2, 3, 7, 12, 22, 41, 75, 138, 254, 467, ...

See also

Brown's Criterion, Fibonacci n-Step Number, Fibonacci Number, Hard Hexagon Entropy Constant, Integer Sequence Primes, Pentagonal Icositetrahedron, Snub Cube-Pentagonal Icositetrahedron Compound, Tetranacci Number, Tribonacci Constant

Portions of this entry contributed by Tony Noe

Portions of this entry contributed by Tito Piezas III

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References

Brenner, J. L. "Linear Recurrence Relations." Amer. Math. Monthly 61, 171-173, 1954.Develin, M. "A Complete Categorization of When Generalized Tribonacci Sequences Can Be Avoided by Additive Partitions." Electronic J. Combinatorics 7, No. 1, R53, 1-7, 2000. http://www.combinatorics.org/Volume_7/Abstracts/v7i1r53.html.Dumitriu, I. "On Generalized Tribonacci Sequences and Additive Partitions." Disc. Math. 219, 65-83, 2000.Feinberg, M. "Fibonacci-Tribonacci." Fib. Quart. 1, 71-74, 1963.Hoggatt, V. E. Jr. "Additive Partitions of the Positive Integers." Fib. Quart. 18, 220-226, 1980.Plouffe, S. "The Tribonacci Constant." http://pi.lacim.uqam.ca/piDATA/tribo.txt.Sloane, N. J. A. Sequences A000073/M1074, A000213/M2454, A001590/M0784, A001644/M2625, A058265, A092835, A092836, A100683,and A112305 in "The On-Line Encyclopedia of Integer Sequences."

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Tribonacci Number

Cite this as:

Noe, Tony; Piezas, Tito III; and Weisstein, Eric W. "Tribonacci Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TribonacciNumber.html

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