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Difference between revisions of "Tribonacci number"

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A member of the [[Tribonacci sequence|Tribonacci sequence]]. The formula for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130190/t1301901.png" />th number is given by A. Shannon in [[#References|[a1]]]:
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A member of the [[Tribonacci sequence]]. The formula for the $n$-th number is given by A. Shannon in [[#References|[a1]]]:
 
+
$$
<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/t130190/t1301902.png" /></td> </tr></table>
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T_n = \sum_{m=0}^{[n/2]} \sum_{r=0}^{[n/3]} \binom{ n-m-2r }{ m+r }\binom{ m+r }{ r }
 
+
$$
Binet's formula for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130190/t1301903.png" />th number is given by W. Spickerman in [[#References|[a2]]]:
 
 
 
<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/t130190/t1301904.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/t130190/t1301905.png" /></td> </tr></table>
 
  
 +
Binet's formula for the $n$-th number is given by W. Spickerman in [[#References|[a2]]]:
 +
$$
 +
T_n = \frac{\rho^{n+2}}{ (\rho-\sigma)(\rho-\bar\sigma) } + \frac{\sigma^{n+2}}{ (\sigma-\rho)(\sigma-\bar\sigma) } + \frac{\bar\sigma^{n+2}}{ (\bar\sigma-\rho)(\bar\sigma-\sigma) }
 +
$$
 
where
 
where
 +
$$
 +
\rho = \frac{1}{3}\left({ (19+3\sqrt{33})^{1/3} + (19-3\sqrt{33})^{1/3} +1 }\right)\,,
 +
$$
 +
and
 +
$$
 +
\sigma = \frac{1}{6}\left({ 2-(19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) + \frac{\sqrt3 i}{6}\left({ (19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) \ .
 +
$$
  
<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/t130190/t1301906.png" /></td> </tr></table>
+
====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</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">[a2]</TD> <TD valign="top">  W. Spickerman,  "Binet's formula for the Tribonacci sequence" ''The Fibonacci Quart.'' , '''15''' : 3  (1977)  pp. 268; 275</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/t130190/t1301907.png" /></td> </tr></table>
+
{{TEX|done}}
 
 
<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/t130190/t1301908.png" /></td> </tr></table>
 
 
 
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130190/t1301909.png" /> is the complex conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130190/t13019010.png" />.
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</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">[a2]</TD> <TD valign="top">  W. Spickerman,  "Binet's formula for the Tribonacci sequence"  ''The Fibonacci Quart.'' , '''15''' :  3  (1977)  pp. 268; 275</TD></TR></table>
 

Revision as of 15:55, 21 March 2018

A member of the Tribonacci sequence. The formula for the $n$-th number is given by A. Shannon in [a1]: $$ T_n = \sum_{m=0}^{[n/2]} \sum_{r=0}^{[n/3]} \binom{ n-m-2r }{ m+r }\binom{ m+r }{ r } $$

Binet's formula for the $n$-th number is given by W. Spickerman in [a2]: $$ T_n = \frac{\rho^{n+2}}{ (\rho-\sigma)(\rho-\bar\sigma) } + \frac{\sigma^{n+2}}{ (\sigma-\rho)(\sigma-\bar\sigma) } + \frac{\bar\sigma^{n+2}}{ (\bar\sigma-\rho)(\bar\sigma-\sigma) } $$ where $$ \rho = \frac{1}{3}\left({ (19+3\sqrt{33})^{1/3} + (19-3\sqrt{33})^{1/3} +1 }\right)\,, $$ and $$ \sigma = \frac{1}{6}\left({ 2-(19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) + \frac{\sqrt3 i}{6}\left({ (19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) \ . $$

References

[a1] A. Shannon, "Tribonacci numbers and Pascal's pyramid" The Fibonacci Quart. , 15 : 3 (1977) pp. 268; 275
[a2] W. Spickerman, "Binet's formula for the Tribonacci sequence" The Fibonacci Quart. , 15 : 3 (1977) pp. 268; 275
How to Cite This Entry:
Tribonacci number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tribonacci_number&oldid=42999
This article was adapted from an original article by Krassimir Atanassov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article