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Tribonacci Constant

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The tribonacci constant is ratio to which adjacent tribonacci numbers tend, and is given by

t=(x^3-x^2-x-1)_1
(1)
=1/3(1+RadicalBox[{19, -, 3, {sqrt(, 33, )}}, 3]+RadicalBox[{19, +, 3, {sqrt(, 33, )}}, 3])
(2)
=1.83929...
(3)

(OEIS A058265).

The tribonacci constant satisfies the identities

(t+1)(t-1)^2=2
(4)
t+t^(-3)=2
(5)
((t-1)(t^2+1))/t=2
(6)
((t+1)^2)/(t(t^2+1))=1
(7)

(P. Moses, pers. comm., Feb. 21, 2005).

The tribonacci constant is extremely prominent in the properties of the snub cube.

See also

Hard Hexagon Entropy Constant, Snub Cube, Tetranacci Constant, Tribonacci Number

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References

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 9, 2003.Pegg, E. Jr. "Shattering the Plane with Twelve New Substitution Tilings Using 2, phi, psi, chi, rho." Mar. 7, 2019. https://blog.wolfram.com/2019/03/07/shattering-the-plane-with-twelve-new-substitution-tilings-using-2-phi-psi-chi-rho/.Sloane, N. J. A. Sequence A058265 in "The On-Line Encyclopedia of Integer Sequences."

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Tribonacci Constant

Cite this as:

Weisstein, Eric W. "Tribonacci Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TribonacciConstant.html

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