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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 (list; constant; graph; refs; listen; history; text; internal format)
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^5-x^4-x^3-x^2-x-1 = 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^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

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), 499-512.

V. Shevelev, A property of n-bonacci 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^4-x^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

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Last modified November 27 21:59 EST 2014. Contains 250286 sequences.