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%I #42 Jan 02 2023 12:30:46
%S 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 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 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 Pentanacci constant: decimal expansion of limit of A001591(n+1)/A001591(n).
%C 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 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 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 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 For n>=5, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - _Vladimir Shevelev_, Mar 21 2014
%C Note that the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - _Bernard Schott_, May 07 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 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 et al., <a href="http://mathworld.wolfram.com/TetranacciConstant.html">Tetranacci Constant.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentanacciConstant.html">Pentanacci Constant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentanacciNumber.html">Pentanacci Number</a>
%e 1.965948236645485337189937375934401396151327177456861393236934508442...
%t RealDigits[Root[x^5-Total[x^Range[0,4]],1],10,120][[1]] (* _Harvey P. Dale_, Mar 22 2017 *)
%o (PARI) solve(x=1, 2, 1+x+x^2+x^3+x^4-x^5) \\ _Michel Marcus_, Mar 21 2014
%Y Cf. A001591.
%Y k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), A086088 (tetranacci), this sequence (pentanacci), A118427 (hexanacci), A118428 (heptanacci).
%K nonn,cons,easy
%O 1,2
%A _Jonathan Vos Post_, Mar 29 2005
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