

A058265


Decimal expansion of the tribonacci constant t, the real root of x^3  x^2  x  1.


94



1, 8, 3, 9, 2, 8, 6, 7, 5, 5, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 5, 1, 8, 5, 2, 5, 6, 4, 6, 5, 3, 2, 8, 6, 6, 0, 0, 4, 2, 4, 1, 7, 8, 7, 4, 6, 0, 9, 7, 5, 9, 2, 2, 4, 6, 7, 7, 8, 7, 5, 8, 6, 3, 9, 4, 0, 4, 2, 0, 3, 2, 2, 2, 0, 8, 1, 9, 6, 6, 4, 2, 5, 7, 3, 8, 4, 3, 5, 4, 1, 9, 4, 2, 8, 3, 0, 7, 0, 1, 4
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OFFSET

1,2


COMMENTS

"The tribonacci constant, the only real solution to the equation x^3  x^2  x  1 = 0, which is related to tribonacci sequences (in which U_n = U_n1 + U_n2 + U_n3) as the Golden Ratio is related to the Fibonacci sequence and its generalizations. This ratio also appears when a snub cube is inscribed in an octahedron or a cube, by analogy once again with the appearance of the Golden Ratio when an icosahedron is inscribed in an octahedron. [John Sharp, 1997]"
The tribonacci constant corresponds to the Golden Section in a tripartite division 1 = u_1 + u_2 + u_3 of a unit line segment; i.e., if 1/u_1 = u_1/u_2 = u_2/u_3 = c, c is the tribonacci constant.  Seppo Mustonen, Apr 19 2005
The other two polynomial roots are the complexconjugated pair 0.4196433776070805662759262... + i* 0.60629072920719936925934...  R. J. Mathar, Oct 25 2008
For n >= 3, round(q^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link.  Vladimir Shevelev, Mar 21 2014
Concerning orthogonal projections, the tribonacci constant is the ratio of the diagonal of a square to the width of a rhombus projected by rotating a square along its diagonal in 3D until the angle of rotation equals the apparent apex angle at approximately 57.065 degrees (also the corresponding angle in the formula generating A256099). See illustration in the links.  Peter M. Chema, Jan 02 2017
Real eigenvalue t of the tribonacci Qmatrix <<1, 1, 1>,<1, 0, 0>,<0, 1, 0>>.
Limit_{n > oo} T(n+1)/T(n) = t (from the T recurrence), where T = {A000073(n+2)}_{n >= 0}. (End)
The nonnegative powers of t are t^n = T(n)*t^2 + (T(n1) + T(n2))*t + T(n1)*1, for n >= 0, with T(n) = A000073(n), with T(1) = 1 and T(2) = 1, This follows from the recurrences derived from t^3 = t^2 + t + 1. See the examples below. For the negative powers see A319200.  Wolfdieter Lang, Oct 23 2018
Note that we have: t + t^(3) = 2, and the knacci constant approaches 2 when k approaches infinity (Martin Gardner).  Bernard Schott, May 16 2022
The roots of this cubic are found from those of y^3  (4/3)*y  38/27, after adding 1/3.  Wolfdieter Lang, Aug 24 2022
The algebraic number t  1 has minimal polynomial x^3 + 2*x^2  2 over Q. The roots coincide with those of y^3  (4/3)*y  38/27, after subtracting 2/3.  Wolfdieter Lang, Sep 20 2022
The value of the ratio R/r of the radius R of a uniform ball to the radius r of a spherical hole in it with a common point of contact, such that the center of gravity of the object lies on the surface of the spherical hole (Schmidt, 2002).  Amiram Eldar, May 20 2023


REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Martin Gardner, The Second Scientific American Book of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.


LINKS

Herbert C. H. Schmidt, Problem 2670, Crux Mathematicorum, Vol. 28, No. 7 (2002), pp. 464465.


FORMULA

t = (1/3)*(1+(19+3*sqrt(33))^(1/3)+(193*sqrt(33))^(1/3)).  Zak Seidov, Jun 08 2005


EXAMPLE

1.8392867552141611325518525646532866004241787460975922467787586394042032220\
81966425738435419428307014141979826859240974164178450746507436943831545\
820499513796249655539644613666121540277972678118941041...
The coefficients of t^2, t, 1 for t^n begin, for n >= 0:
n t^2 t 1

0 0 0 1
1 0 1 0
2 1 0 0
1 1 1 1
4 2 2 1
5 4 3 2
6 7 6 4
7 13 11 7
8 24 20 13
9 44 37 24
10 81 68 44
... (End)


MAPLE



MATHEMATICA

RealDigits[ N[ 1/3 + 1/3*(19  3*Sqrt[33])^(1/3) + 1/3*(19 + 3*Sqrt[33])^(1/3), 100]] [[1]]
RealDigits[Root[x^3x^2x1, 1], 10, 120][[1]] (* Harvey P. Dale, Mar 23 2019 *)


PROG

(PARI) default(realprecision, 20080); x=solve(x=1, 2, x^3  x^2  x  1); for (n=1, 20000, d=floor(x); x=(xd)*10; write("b058265.txt", n, " ", d)); \\ Harry J. Smith, May 30 2009
(PARI) q=(1+sqrtn(19+3*sqrt(33), 3)+sqrtn(193*sqrt(33), 3))/3 \\ Use \p# to set 'realprecision'.  M. F. Hasler, Mar 23 2014
(Maxima) set_display(none)$ fpprec:100$ bfloat(rhs(solve(t^3t^2t1, t)[3])); /* Dimitri Papadopoulos, Nov 09 2023 */


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



