%I #200 Nov 12 2023 12:58:06
%S 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,
%T 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,
%U 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
%N Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1.
%C "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_n-1 + U_n-2 + U_n-3) 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]"
%C 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
%C The other two polynomial roots are the complex-conjugated pair -0.4196433776070805662759262... +- i* 0.60629072920719936925934... - _R. J. Mathar_, Oct 25 2008
%C For n >= 3, round(q^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - _Vladimir Shevelev_, Mar 21 2014
%C 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
%C From _Wolfdieter Lang_, Aug 10 2018: (Start)
%C Real eigenvalue t of the tribonacci Q-matrix <<1, 1, 1>,<1, 0, 0>,<0, 1, 0>>.
%C Limit_{n -> oo} T(n+1)/T(n) = t (from the T recurrence), where T = {A000073(n+2)}_{n >= 0}. (End)
%C The nonnegative powers of t are t^n = T(n)*t^2 + (T(n-1) + T(n-2))*t + T(n-1)*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
%C Note that we have: t + t^(-3) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - _Bernard Schott_, May 16 2022
%C 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
%C 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
%C 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
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
%D Martin Gardner, The Second Scientific American Book of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.
%H Harry J. Smith, <a href="/A058265/b058265.txt">Table of n, a(n) for n = 1..20000</a>
%H A. Beha et al., <a href="http://www.jstor.org/stable/30037493">The convergence of diffy boxes</a>, American Mathematical Monthly, Vol. 112 (2005), pp. 426-439.
%H F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://doi.org/10.37236/8905">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), Article P1.52.
%H O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, <a href="http://dx.doi.org/10.5539/jmr.v7n2p34">The Cyclic Groups via Bezout Matrices</a>, Journal of Mathematics Research, Vol. 7, No. 2 (2015), pp. 34-41.
%H Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, <a href="https://doi.org/10.7546/nntdm.2020.26.1.179-190">On the Generalized Fibonacci-circulant-Hurwitz numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 26, No. 1 (2020), 179-190.
%H Peter M. Chema, <a href="/A058265/a058265_2.pdf">Tribonacci constant as ratio of square to rhombus projection</a>.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
%H S. Litsyn and Vladimir Shevelev, <a href="http://dx.doi.org/10.1142/S1793042105000339">Irrational Factors Satisfying the Little Fermat Theorem</a>, International Journal of Number Theory, Vol. 1, No. 4 (2005), 499-512.
%H Xerardo Neira, <a href="/A058265/a058265_3.pdf">A geometric construction of the tribonacci constant with marked ruler and compass</a>.
%H Tito Piezas III, <a href="https://sites.google.com/view/tpiezas/0012-article-2-tribonacci-constant-and-pi">Tribonacci constant and Pi</a>.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/tribo.txt">Tribonacci constant to 2000 digits</a>.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap89.html">The Tribonacci constant(to 1000 digits)</a>.
%H Herbert C. H. Schmidt, <a href="https://cms.math.ca/publications/crux/issue/?volume=28&issue=7">Problem 2670</a>, Crux Mathematicorum, Vol. 28, No. 7 (2002), pp. 464-465.
%H Vladimir Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2014-March/012750.html">A property of n-bonacci constant</a>, Seqfan (Mar 23 2014).
%H Nikita Sidorov, <a href="https://doi.org/10.1016/j.jnt.2008.11.003">Expansions in non-integer bases: Lower, middle and top orders</a>, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Lemma 4.1 p. 750.
%H Kees van Prooijen, <a href="http://www.kees.cc/gldsec.html">The Odd Golden Section</a>.
%H Kees van Prooijen, <a href="/A058265/a058265.jpg">Tribonacci Box (analog of Golden Rectangle)</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TribonacciConstant.html">Tribonacci Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>.
%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F t = (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)). - _Zak Seidov_, Jun 08 2005
%F t = 1 - Sum_{k>=1} A057597(k+2)/(T_k*T_(k+1)), where T_n = A000073(n+1). - _Vladimir Shevelev_, Mar 02 2013
%F 1/t + 1/t^2 + 1/t^3 = 1/A058265 + 1/A276800 + 1/A276801 = 1. - _N. J. A. Sloane_, Oct 28 2016
%F t = (4/3)*cosh((1/3)*arccosh(19/8)) + 1/3. - _Wolfdieter Lang_, Aug 24 2022
%e 1.8392867552141611325518525646532866004241787460975922467787586394042032220\
%e 81966425738435419428307014141979826859240974164178450746507436943831545\
%e 820499513796249655539644613666121540277972678118941041...
%e From _Wolfdieter Lang_, Oct 23 2018: (Start)
%e The coefficients of t^2, t, 1 for t^n begin, for n >= 0:
%e n t^2 t 1
%e -------------------
%e 0 0 0 1
%e 1 0 1 0
%e 2 1 0 0
%e 1 1 1 1
%e 4 2 2 1
%e 5 4 3 2
%e 6 7 6 4
%e 7 13 11 7
%e 8 24 20 13
%e 9 44 37 24
%e 10 81 68 44
%e ... (End)
%p Digits:=200; fsolve(x^3=x^2+x+1); # _N. J. A. Sloane_, Mar 16 2019
%t RealDigits[ N[ 1/3 + 1/3*(19 - 3*Sqrt[33])^(1/3) + 1/3*(19 + 3*Sqrt[33])^(1/3), 100]] [[1]]
%t RealDigits[Root[x^3-x^2-x-1,1],10,120][[1]] (* _Harvey P. Dale_, Mar 23 2019 *)
%o (PARI) default(realprecision, 20080); x=solve(x=1, 2, x^3 - x^2 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b058265.txt", n, " ", d)); \\ _Harry J. Smith_, May 30 2009
%o (PARI) q=(1+sqrtn(19+3*sqrt(33),3)+sqrtn(19-3*sqrt(33),3))/3 \\ Use \p# to set 'realprecision'. - _M. F. Hasler_, Mar 23 2014
%o (Maxima) set_display(none)$ fpprec:100$ bfloat(rhs(solve(t^3-t^2-t-1,t)[3])); /* _Dimitri Papadopoulos_, Nov 09 2023 */
%Y Cf. A000073, A019712 (continued fraction), A133400, A254231, A158919 (spectrum = floor(n*t)), A357101 (x^3-2*x^2-2).
%Y Cf. A192918 (reciprocal), A276800 (square), A276801 (cube), A319200.
%Y k-nacci constants: A001622 (Fibonacci), this sequence (tribonacci), A086088 (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).
%K nonn,cons
%O 1,2
%A _Robert G. Wilson v_, Dec 07 2000