|
%I #63 Oct 27 2023 10:03:03
%S 1,9,2,7,5,6,1,9,7,5,4,8,2,9,2,5,3,0,4,2,6,1,9,0,5,8,6,1,7,3,6,6,2,2,
%T 1,6,8,6,9,8,5,5,4,2,5,5,1,6,3,3,8,4,7,2,7,1,4,6,6,4,7,0,3,8,0,0,9,6,
%U 6,6,0,6,2,2,9,7,8,1,5,5,5,9,1,4,9,8,1,8,2,5,3,4,6,1,8,9,0,6,5,3,2,5
%N Decimal expansion of the limit of the ratio of consecutive terms in the tetranacci sequence A000078.
%C The tetranacci constant corresponds to the Golden Section in a quadripartite division 1 = u_1 + u_2 + u_3 + u_4 of a unit line segment, i.e., if 1/u_1 = u_1/u_2 = u_2/u_3 = u_3/u_4 = c, c is the tetranacci constant. - _Seppo Mustonen_, Apr 19 2005
%C The other 3 polynomial roots of 1+x+x^2+x^3-x^4 are -0.77480411321543385... and the complex-conjugated pair -0.07637893113374572508475 +- i * 0.814703647170386526841... - _R. J. Mathar_, Oct 25 2008
%C The continued fraction expansion starts 1, 1, 12, 1, 4, 7, 1, 21, 1, 2, 1, 4, 6, 1, 10, 1, 2, 2, 1, 7, 1, 1,... - _R. J. Mathar_, Mar 09 2012
%C For n>=4, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - _Vladimir Shevelev_, Mar 21 2014
%C Note that we have: c + c^(-4) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - _Bernard Schott_, May 09 2022
%D Martin Gardner, The Second Scientific American Book Of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.
%H 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 (2020) Vol. 26, No. 1, 179-190.
%H 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 Gültekin, İnci; Deveci, Ömür, <a href="https://doi.org/10.1515/math-2016-0100">On the arrowhead-Fibonacci numbers</a>. Open Math. 14, 1104-1113 (2016).
%H 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 Vladimir Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2014-March/012750.html">A property of n-bonacci constant</a>, Seqfan (Mar 23 2014)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetranacciNumber.html">Tetranacci Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiskCoveringProblem.html">Disk Covering Problem</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetranacciConstant.html">Tetranacci Constant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F Equals 1/4 + sqrt(11/48 - s/72 + 7/s) + sqrt(11/24 + s/72 - 7/s + 1 / sqrt(704/507 - 128 * s/1521 + 7168 / (169 * s))) where s = (sqrt(177304464) + 7020)^(1/3). - _Michal Paulovic_, Oct 08 2022
%e 1.927561975...
%t RealDigits[Root[ -1-#1-#1^2-#1^3+#1^4&, 2], 10, 110][[1]]
%o (PARI) real(polroots(1+x+x^2+x^3-x^4)[2]) \\ _Charles R Greathouse IV_, Jul 19 2012
%o (PARI) polrootsreal(1+x+x^2+x^3-x^4)[2] \\ _Charles R Greathouse IV_, Apr 14 2014
%Y Cf. A000078.
%Y k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), this sequence (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Jul 08 2003
|
