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Difference between revisions of "Tribonacci sequence"

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An extension of the sequence of [[Fibonacci numbers|Fibonacci numbers]] having the form (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302001.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302003.png" /> given constants):
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An extension of the sequence of [[Fibonacci numbers|Fibonacci numbers]] having the form (with $a$, $b$, $c$ given constants):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302004.png" /></td> </tr></table>
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$$t_0=a,t_1=b,t_2=c,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302005.png" /></td> </tr></table>
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$$t_{n+3}=t_{n+2}+t_{n+1}+t_n(n\geq0).$$
  
The concept was introduced by the fourteen-year-old student M. Feinberg in 1963 in [[#References|[a1]]] for the case: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302007.png" />. The basic properties are introduced in [[#References|[a5]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a2]]].
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The concept was introduced by the fourteen-year-old student M. Feinberg in 1963 in [[#References|[a1]]] for the case: $a=b=1$, $c=2$. The basic properties are introduced in [[#References|[a5]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a2]]].
  
 
The Tribonacci sequence was generalized in [[#References|[a6]]], [[#References|[a7]]] to the form of two sequences:
 
The Tribonacci sequence was generalized in [[#References|[a6]]], [[#References|[a7]]] to the form of two sequences:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302008.png" /></td> </tr></table>
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$$a_{n+3}=u_{n+2}+w_{n+1}+y_n,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t1302009.png" /></td> </tr></table>
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$$b_{n+3}=v_{n+2}+x_{n+1}+z_n,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t13020010.png" /> and each of the tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t13020011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t13020012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t13020013.png" /> contains the two symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t13020014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130200/t13020015.png" />. There are eight different such schemes. An open problem (as of 2000) is the construction of an explicit formula for each of them.
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where $u,v,w,x,y,z\in\{a,b\}$ and each of the tuples $(u,v)$, $(w,x)$, $(y,z)$ contains the two symbols $a$ and $b$. There are eight different such schemes. An open problem (as of 2000) is the construction of an explicit formula for each of them.
  
 
See also [[Tribonacci number|Tribonacci number]].
 
See also [[Tribonacci number|Tribonacci number]].

Latest revision as of 19:40, 31 July 2014

An extension of the sequence of Fibonacci numbers having the form (with $a$, $b$, $c$ given constants):

$$t_0=a,t_1=b,t_2=c,$$

$$t_{n+3}=t_{n+2}+t_{n+1}+t_n(n\geq0).$$

The concept was introduced by the fourteen-year-old student M. Feinberg in 1963 in [a1] for the case: $a=b=1$, $c=2$. The basic properties are introduced in [a5], [a3], [a4], [a2].

The Tribonacci sequence was generalized in [a6], [a7] to the form of two sequences:

$$a_{n+3}=u_{n+2}+w_{n+1}+y_n,$$

$$b_{n+3}=v_{n+2}+x_{n+1}+z_n,$$

where $u,v,w,x,y,z\in\{a,b\}$ and each of the tuples $(u,v)$, $(w,x)$, $(y,z)$ contains the two symbols $a$ and $b$. There are eight different such schemes. An open problem (as of 2000) is the construction of an explicit formula for each of them.

See also Tribonacci number.

References

[a1] M. Feinberg, "Fibonacci–Tribonacci" The Fibonacci Quart. , 1 : 3 (1963) pp. 71–74
[a2] C. Valavigi, "Properties of Tribonacci numbers" The Fibonacci Quart. , 10 : 3 (1972) pp. 231–246
[a3] A. Scott, T. Delaney, V. Hoggatt Jr., "The Tribonacci sequence" The Fibonacci Quart. , 15 : 3 (1977) pp. 193–200
[a4] A. Shannon, "Tribonacci numbers and Pascal's pyramid" The Fibonacci Quart. , 15 : 3 (1977) pp. 268; 275
[a5] I. Bruce, "A modified Tribonacci sequence" The Fibonacci Quart. , 22 : 3 (1984) pp. 244–246
[a6] K. Atanassov, J. Hlebarova, S. Mihov, "Recurrent formulas of the generalized Fibonacci and Tribonacci sequences" The Fibonacci Quart. , 30 : 1 (1992) pp. 77–79
[a7] J.-Z. Lee, J.-S. Lee, "Some properties of the generalization of the Fibonacci sequence" The Fibonacci Quart. , 25 : 2 (1987) pp. 111–117
How to Cite This Entry:
Tribonacci sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tribonacci_sequence&oldid=32619
This article was adapted from an original article by Krassimir Atanassov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article