

A103814


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


9



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, 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, 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
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OFFSET

1,2


COMMENTS

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^5x^4x^3x^2x1 = 0. Note that we have: P + P^5 = 2.
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
The other 4 roots of the polynomial 1+x+x^2+x^3+x^4x^5 are the two complexconjugated pairs 0.6783507129699967... + i * 0.458536187273144499.. and 0.1953765946472540452... + i * 0.848853640546245551858...  R. J. Mathar, Oct 25 2008
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
For n>=5, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link.  Vladimir Shevelev, Mar 21 2014


LINKS

Table of n, a(n) for n=1..105.
S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499512.
V. Shevelev, A property of nbonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein et al., Tetranacci Constant.
Eric Weisstein's World of Mathematics, Pentanacci Constant
Eric Weisstein's World of Mathematics, Pentanacci Number


EXAMPLE

1.965948236645485337189937375934401396151327177456861393236934508442...


PROG

(PARI) solve(x=1, 2, 1+x+x^2+x^3+x^4x^5) \\ Michel Marcus, Mar 21 2014


CROSSREFS

Cf. A001591, A058265, A086088.
Sequence in context: A102047 A144665 A019884 * A245770 A117020 A011459
Adjacent sequences: A103811 A103812 A103813 * A103815 A103816 A103817


KEYWORD

nonn,cons,easy


AUTHOR

Jonathan Vos Post, Mar 29 2005


STATUS

approved



