|
| |
|
|
A058265
|
|
Decimal expansion of the tribonacci constant, the solution to x^3=x^2+x+1.
|
|
26
|
|
|
|
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... [From R. J. Mathar, Oct 25 2008]
|
|
|
REFERENCES
|
A. Beha et al., The convergence of diffy boxes, American Mathematical Monthly, Vol. 112 (2005), pp. 426-439.
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
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)
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]
|
|
|
CROSSREFS
|
Cf. A019712.
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
|
| |
|
|
