View source for Tribonacci sequence

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):

<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>

<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>

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]]].

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>

<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>

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.

See also [[Tribonacci number|Tribonacci number]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Feinberg,   "Fibonacci–Tribonacci"  ''The Fibonacci Quart.'' , '''1''' :  3  (1963)  pp. 71–74</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Valavigi,   "Properties of Tribonacci numbers"  ''The Fibonacci Quart.'' , '''10''' :  3  (1972)  pp. 231–246</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Scott,   T. Delaney,   V. Hoggatt Jr.,   "The Tribonacci sequence"  ''The Fibonacci Quart.'' , '''15''' :  3  (1977)  pp. 193–200</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Shannon,   "Tribonacci numbers and Pascal's pyramid"  ''The Fibonacci Quart.'' , '''15''' :  3  (1977)  pp. 268; 275</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Bruce,   "A modified Tribonacci sequence"  ''The Fibonacci Quart.'' , '''22''' :  3  (1984)  pp. 244–246</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Atanassov,   J. Hlebarova,   S. Mihov,   "Recurrent formulas of the generalized Fibonacci and Tribonacci sequences"  ''The Fibonacci Quart.'' , '''30''' :  1  (1992)  pp. 77–79</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.-Z. Lee,   J.-S. Lee,   "Some properties of the generalization of the Fibonacci sequence"  ''The Fibonacci Quart.'' , '''25''' :  2  (1987)  pp. 111–117</TD></TR></table>