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A118427 Decimal expansion of hexanacci constant. 7

%I #48 Jan 02 2023 12:30:46

%S 1,9,8,3,5,8,2,8,4,3,4,2,4,3,2,6,3,3,0,3,8,5,6,2,9,2,9,3,3,9,1,4,2,5,

%T 7,5,2,7,3,0,0,8,0,8,6,5,5,6,8,8,2,1,7,5,3,2,1,6,3,5,9,0,6,5,6,5,6,7,

%U 0,2,2,7,8,0,1,4,1,7,2,4,0,2,9,8,6,5,7,5,0,7,0,2,2,6,8,9,9,7,9,7,3,2,7,7,5

%N Decimal expansion of hexanacci constant.

%C The continued fraction expansion starts 1, 1, 59, 1, 10, 2, 1, 6, 2, 1, 6, 1, 1, 7, 1, 71, 7, 1, 6, 8, ... - _R. J. Mathar_, Mar 09 2012

%C For n>=7, 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^(-6) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - _Bernard Schott_, May 06 2022

%D Martin Gardner, The Second Scientific American Book Of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, 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's World of Mathematics, <a href="http://mathworld.wolfram.com/HexanacciNumber.html">Hexanacci Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexanacciConstant.html">Hexanacci Constant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexanacciNumber.html">Hexanacci Number</a>

%e 1.9835828434243263303...

%t RealDigits[ Root[ x^6 - x^5 - x^4 - x^3 - x^2 - x - 1, 2] , 10, 105] // First (* _Jean-François Alcover_, Feb 07 2013 *)

%Y Cf. A001592.

%Y k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), A086088 (tetranacci), A103814 (pentanacci), this sequence (hexanacci), A118428 (heptanacci).

%K nonn,cons

%O 1,2

%A _Eric W. Weisstein_, Apr 27 2006

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