

A058265


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


54



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


REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
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. 426439.
O. Deveci, Y. Akuzum, E. Karaduman, O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 3441.
Peter M. Chema, Tribonacci constant as ratio of square to rhombus projection
S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499512.
Tito Piezas III, Tribonacci constant and Pi
Simon Plouffe, Tribonacci constant to 2000 digits
Simon Plouffe, The Tribonacci constant(to 1000 digits)
Kees van Prooijen, The Odd Golden Section
Kees van Prooijen, Tribonacci Box (analog of Golden Rectangle)
V. Shevelev, A property of nbonacci constant, Seqfan (Mar 23 2014).
Nikita Sidorov, Expansions in noninteger bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741754. See Lemma 4.1 p. 750.
Eric Weisstein's World of Mathematics, Tribonacci Number
Eric Weisstein's World of Mathematics, Tribonacci Constant
Eric Weisstein's World of Mathematics, Fibonacci nStep Number


FORMULA

t = (1/3)*(1+(19+3*sqrt(33))^(1/3)+(193*sqrt(33))^(1/3)) = 1.8392867552141611325518525646532866004241...  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


EXAMPLE

1.8392867552141611325518525646532866004241787460975922467787586394042032220\
81966425738435419428307014141979826859240974164178450746507436943831545\
820499513796249655539644613666121540277972678118941041...


MATHEMATICA

RealDigits[ N[ 1/3 + 1/3*(19  3*Sqrt[33])^(1/3) + 1/3*(19 + 3*Sqrt[33])^(1/3), 100]] [[1]]


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


CROSSREFS

Cf. A000073, A019712 (continued fraction), A133400, A254231, A158919 (spectrum = floor(n*t)).
Cf. also A276800, A276801.
Sequence in context: A146482 A019938 A170937 * A135005 A090734 A200614
Adjacent sequences: A058262 A058263 A058264 * A058266 A058267 A058268


KEYWORD

nonn,cons


AUTHOR

Robert G. Wilson v, Dec 07 2000


STATUS

approved



