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A058265 Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1. 92
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 (list; constant; graph; refs; listen; history; text; internal format)
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_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]"

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 complex-conjugated 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

From Wolfdieter Lang, Aug 10 2018: (Start)

Real eigenvalue t of the tribonacci Q-matrix <<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(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

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

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

REFERENCES

S. 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

Harry J. Smith, Table of n, a(n) for n = 1..20000

A. Beha et al., The convergence of diffy boxes, American Mathematical Monthly, Vol. 112 (2005), pp. 426-439.

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41.

Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.

Peter M. Chema, Tribonacci constant as ratio of square to rhombus projection

Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.

Xerardo Neira, A geometric construction of the tribonacci constant with marked ruler and compass

Tito Piezas III, Tribonacci constant and Pi

Simon Plouffe, Tribonacci constant to 2000 digits

Simon Plouffe, The Tribonacci constant(to 1000 digits)

Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014).

Nikita Sidorov, Expansions in non-integer bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Lemma 4.1 p. 750.

Kees van Prooijen, The Odd Golden Section

Kees van Prooijen, Tribonacci Box (analog of Golden Rectangle)

Eric Weisstein's World of Mathematics, Tribonacci Number

Eric Weisstein's World of Mathematics, Tribonacci Constant

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

FORMULA

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

t = 1 - Sum_{k>=1} A057597(k+2)/(T_k*T_(k+1)), where T_n = A000073(n+1). - Vladimir Shevelev, Mar 02 2013

1/t + 1/t^2 + 1/t^3 = 1/A058265 + 1/A276800 + 1/A276801 = 1. - N. J. A. Sloane, Oct 28 2016

t = (4/3)*cosh((1/3)*arccosh(19/8)) + 1/3. - Wolfdieter Lang, Aug 24 2022

EXAMPLE

1.8392867552141611325518525646532866004241787460975922467787586394042032220\

81966425738435419428307014141979826859240974164178450746507436943831545\

820499513796249655539644613666121540277972678118941041...

From Wolfdieter Lang, Oct 23 2018: (Start)

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

Digits:=200; fsolve(x^3=x^2+x+1); # N. J. A. Sloane, Mar 16 2019

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^3-x^2-x-1, 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=(x-d)*10; write("b058265.txt", n, " ", d)); } \\ Harry J. Smith, May 30 2009

(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

CROSSREFS

Cf. A000073, A019712 (continued fraction), A133400, A254231, A158919 (spectrum = floor(n*t)), A357101 (x^3-2*x^2-2).

Cf. A192918 (reciprocal), A276800 (square), A276801 (cube), A319200.

k-nacci constants: A001622 (Fibonacci), this sequence (tribonacci), A086088 (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).

Sequence in context: A146482 A019938 A170937 * A357528 A135005 A090734

Adjacent sequences: A058262 A058263 A058264 * A058266 A058267 A058268

KEYWORD

nonn,cons

AUTHOR

Robert G. Wilson v, Dec 07 2000

STATUS

approved

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Last modified November 30 04:37 EST 2022. Contains 358431 sequences. (Running on oeis4.)