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A058265 Decimal expansion of the tribonacci constant, the solution to x^3=x^2+x+1. 29
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

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. 426-439.

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

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 n-bonacci constant, Seqfan (Mar 23 2014)

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

q = (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)) = 1.8392867552141611325518525646532866004241... - Zak Seidov, Jun 08 2005

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

EXAMPLE

1.839286755214161...

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=(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. A019712.

Sequence in context: A146482 A019938 A170937 * A135005 A090734 A200614

Adjacent sequences:  A058262 A058263 A058264 * A058266 A058267 A058268

KEYWORD

nonn,cons,changed

AUTHOR

Robert G. Wilson v, Dec 07 2000

STATUS

approved

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Last modified April 24 00:51 EDT 2014. Contains 240947 sequences.