Difference between revisions of "Tribonacci number"
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| − | A member of the [[ | + | A member of the [[Tribonacci sequence]]. The formula for the $n$-th number is given by A. Shannon in [[#References|[a1]]]: |
| − | + | $$ | |
| − | + | T_n = \sum_{m=0}^{[n/2]} \sum_{r=0}^{[n/3]} \binom{ n-m-2r }{ m+r }\binom{ m+r }{ r } | |
| − | + | $$ | |
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| + | 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 | + | ====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 {{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.'' , '''20''' (1981) pp. 118–120 {{ZBL|0486.10011}}</TD></TR> | ||
| + | </table> | ||
| − | + | {{TEX|done}} | |
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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=17204
Tribonacci number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tribonacci_number&oldid=17204
This article was adapted from an original article by Krassimir Atanassov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article