cp's OEIS Frontend

A103814 Pentanacci constant: decimal expansion of limit of A001591(n+1)/A001591(n).

%I A103814 #46 Feb 16 2025 08:32:56
%S A103814 1,9,6,5,9,4,8,2,3,6,6,4,5,4,8,5,3,3,7,1,8,9,9,3,7,3,7,5,9,3,4,4,0,1,
%T A103814 3,9,6,1,5,1,3,2,7,1,7,7,4,5,6,8,6,1,3,9,3,2,3,6,9,3,4,5,0,8,4,4,2,2,
%U A103814 5,2,7,1,2,8,7,1,8,8,6,8,8,1,7,3,4,8,1,8,6,6,5,5,5,4,6,3,0,4,7,2,0,2,1,3,0
%N A103814 Pentanacci constant: decimal expansion of limit of A001591(n+1)/A001591(n).
%C A103814 The pentanacci constant P is the limit as n -> infinity of the ratio of Pentanacci(n+1)/Pentanacci(n) = A001591(n+1)/A001591(n), which is the principal root of x^5-x^4-x^3-x^2-x-1 = 0. Note that we have: P + P^-5 = 2.
%C A103814 The pentanacci constant corresponds to the Golden Section in a fivepartite division 1 = u_1 + u_2 + u_3 + u_4 + u_5 of a unit line segment, i.e., if 1/u_1 = u_1/u_2 = u_2/u_3 = u_3/u_4 + u_4/u_5 = c, c is the pentanacci constant. - _Seppo Mustonen_, Apr 19 2005
%C A103814 The other 4 roots of the polynomial 1+x+x^2+x^3+x^4-x^5 are the two complex-conjugated pairs -0.6783507129699967... +- i * 0.458536187273144499.. and 0.1953765946472540452... +- i * 0.848853640546245551858... - _R. J. Mathar_, Oct 25 2008
%C A103814 The continued fraction expansion is 1, 1, 28, 2, 1, 2, 1, 1, 1, 2, 4, 2, 1, 3, 1, 6, 1, 4, 1, 1, 5, 3, 2, 15, 69, 1, 1, 14, 1, 8, 1, 6,... - _R. J. Mathar_, Mar 09 2012
%C A103814 For n>=5, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - _Vladimir Shevelev_, Mar 21 2014
%C A103814 Note that the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - _Bernard Schott_, May 07 2022
%D A103814 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 A103814 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 A103814 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 A103814 Eric Weisstein et al., <a href="https://mathworld.wolfram.com/TetranacciConstant.html">Tetranacci Constant.</a>
%H A103814 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentanacciConstant.html">Pentanacci Constant</a>
%H A103814 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentanacciNumber.html">Pentanacci Number</a>
%H A103814 <a href="/index/Al#algebraic_05">Index entries for algebraic numbers, degree 5</a>.
%e A103814 1.965948236645485337189937375934401396151327177456861393236934508442...
%t A103814 RealDigits[Root[x^5-Total[x^Range[0,4]],1],10,120][[1]] (* _Harvey P. Dale_, Mar 22 2017 *)
%o A103814 (PARI) solve(x=1, 2, 1+x+x^2+x^3+x^4-x^5) \\ _Michel Marcus_, Mar 21 2014
%Y A103814 Cf. A001591.
%Y A103814 k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), A086088 (tetranacci), this sequence (pentanacci), A118427 (hexanacci), A118428 (heptanacci).
%K A103814 nonn,cons,easy
%O A103814 1,2
%A A103814 _Jonathan Vos Post_, Mar 29 2005