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Difference between revisions of "Tribonacci number"

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<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>
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<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  {{ZBL|0385.05006}}</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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Spickerman,  "Binet's formula for the Tribonacci sequence"  ''The Fibonacci Quart.'' , '''20'''  (1981)  pp. 118–120  {{ZBL|0486.10011}}</TD></TR>
 
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Latest revision as of 20:08, 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 Zbl 0385.05006
[a2] W. Spickerman, "Binet's formula for the Tribonacci sequence" The Fibonacci Quart. , 20 (1981) pp. 118–120 Zbl 0486.10011
How to Cite This Entry:
Tribonacci number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tribonacci_number&oldid=43000
This article was adapted from an original article by Krassimir Atanassov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article