This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A058265 #205 Feb 16 2025 08:32:43
%S A058265 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 A058265 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 A058265 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 A058265 Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1.
%C A058265 "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 A058265 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 A058265 The other two polynomial roots are the complex-conjugated pair -0.4196433776070805662759262... +- i* 0.60629072920719936925934... - _R. J. Mathar_, Oct 25 2008
%C A058265 For n >= 3, round(q^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - _Vladimir Shevelev_, Mar 21 2014
%C A058265 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 A058265 From _Wolfdieter Lang_, Aug 10 2018: (Start)
%C A058265 Real eigenvalue t of the tribonacci Q-matrix <<1, 1, 1>,<1, 0, 0>,<0, 1, 0>>.
%C A058265 Limit_{n -> oo} T(n+1)/T(n) = t (from the T recurrence), where T = {A000073(n+2)}_{n >= 0}. (End)
%C A058265 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 A058265 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 A058265 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 A058265 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 A058265 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 A058265 Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
%D A058265 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 A058265 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.
%H A058265 Harry J. Smith, <a href="/A058265/b058265.txt">Table of n, a(n) for n = 1..20000</a>
%H A058265 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 A058265 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 A058265 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 A058265 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 A058265 Peter M. Chema, <a href="/A058265/a058265_2.pdf">Tribonacci constant as ratio of square to rhombus projection</a>.
%H A058265 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 A058265 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 A058265 Xerardo Neira, <a href="/A058265/a058265_3.pdf">A geometric construction of the tribonacci constant with marked ruler and compass</a>.
%H A058265 Tito Piezas III, <a href="https://sites.google.com/view/tpiezas/0012-article-2-tribonacci-constant-and-pi">Tribonacci constant and Pi</a>.
%H A058265 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/tribo.txt">Tribonacci constant to 2000 digits</a>.
%H A058265 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap89.html">The Tribonacci constant(to 1000 digits)</a>.
%H A058265 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 A058265 Vladimir Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-March/012750.html">A property of n-bonacci constant</a>, Seqfan (Mar 23 2014).
%H A058265 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 A058265 Kees van Prooijen, <a href="http://www.kees.cc/gldsec.html">The Odd Golden Section</a>.
%H A058265 Kees van Prooijen, <a href="/A058265/a058265.jpg">Tribonacci Box (analog of Golden Rectangle)</a>.
%H A058265 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>.
%H A058265 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TribonacciConstant.html">Tribonacci Constant</a>.
%H A058265 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>.
%H A058265 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A058265 t = (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)). - _Zak Seidov_, Jun 08 2005
%F A058265 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 A058265 1/t + 1/t^2 + 1/t^3 = 1/A058265 + 1/A276800 + 1/A276801 = 1. - _N. J. A. Sloane_, Oct 28 2016
%F A058265 t = (4/3)*cosh((1/3)*arccosh(19/8)) + 1/3. - _Wolfdieter Lang_, Aug 24 2022
%F A058265 t = 2 - Sum_{k>=0} binomial(4*k + 2, k)/((k + 1)*2^(4*k + 3)). - _Antonio Graciá Llorente_, Oct 28 2024
%e A058265 1.8392867552141611325518525646532866004241787460975922467787586394042032220\
%e A058265 81966425738435419428307014141979826859240974164178450746507436943831545\
%e A058265 820499513796249655539644613666121540277972678118941041...
%e A058265 From _Wolfdieter Lang_, Oct 23 2018: (Start)
%e A058265 The coefficients of t^2, t, 1 for t^n begin, for n >= 0:
%e A058265 n t^2 t 1
%e A058265 -------------------
%e A058265 0 0 0 1
%e A058265 1 0 1 0
%e A058265 2 1 0 0
%e A058265 1 1 1 1
%e A058265 4 2 2 1
%e A058265 5 4 3 2
%e A058265 6 7 6 4
%e A058265 7 13 11 7
%e A058265 8 24 20 13
%e A058265 9 44 37 24
%e A058265 10 81 68 44
%e A058265 ... (End)
%p A058265 Digits:=200; fsolve(x^3=x^2+x+1); # _N. J. A. Sloane_, Mar 16 2019
%t A058265 RealDigits[ N[ 1/3 + 1/3*(19 - 3*Sqrt[33])^(1/3) + 1/3*(19 + 3*Sqrt[33])^(1/3), 100]] [[1]]
%t A058265 RealDigits[Root[x^3-x^2-x-1,1],10,120][[1]] (* _Harvey P. Dale_, Mar 23 2019 *)
%o A058265 (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 A058265 (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 A058265 (Maxima) set_display(none)$ fpprec:100$ bfloat(rhs(solve(t^3-t^2-t-1,t)[3])); /* _Dimitri Papadopoulos_, Nov 09 2023 */
%Y A058265 Cf. A000073, A019712 (continued fraction), A133400, A254231, A158919 (spectrum = floor(n*t)), A357101 (x^3-2*x^2-2).
%Y A058265 Cf. A192918 (reciprocal), A276800 (square), A276801 (cube), A319200.
%Y A058265 k-nacci constants: A001622 (Fibonacci), this sequence (tribonacci), A086088 (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).
%K A058265 nonn,cons
%O A058265 1,2
%A A058265 _Robert G. Wilson v_, Dec 07 2000