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

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

Offset: 1

Author

Jonathan Vos Post, Mar 29 2005

Keywords

• nonn
• cons
• easy

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

Note that the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - Bernard Schott, May 07 2022

Examples

			1.965948236645485337189937375934401396151327177456861393236934508442...
		

References

• Martin Gardner, The Second Scientific American Book Of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.

Links

• S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
• Vladimir 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
• Index entries for algebraic numbers, degree 5.

Crossrefs

Cf. A001591.

k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), A086088 (tetranacci), this sequence (pentanacci), A118427 (hexanacci), A118428 (heptanacci).

Programs

Mathematica

RealDigits[Root[x^5-Total[x^Range[0,4]],1],10,120][[1]] (* Harvey P. Dale, Mar 22 2017 *)

PARI

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