A118428 Decimal expansion of heptanacci constant.

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

Offset: 1

Author

Eric W. Weisstein, Apr 27 2006

Keywords

• nonn
• cons

Comments

Other roots of the equation x^7 - x^6 - ... - x - 1 see in A239566. For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - Vladimir Shevelev, Mar 21 2014

Note that we have: c + c^(-7) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - Bernard Schott, May 07 2022

Examples

			1.9919641966050350210...
		

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's World of Mathematics, Heptanacci Number
• Eric Weisstein's World of Mathematics, Heptanacci Constant
• Index entries for algebraic numbers, degree 7.

Crossrefs

Cf. A066178, A239566.

k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), A086088 (tetranacci), A103814 (pentanacci), A118427 (hexanacci), this sequence (heptanacci).

Programs

Mathematica

RealDigits[x/.FindRoot[x^7+Total[-x^Range[0,6]]==0,{x,2}, WorkingPrecision-> 110]][[1]] (* Harvey P. Dale, Dec 13 2011 *)

PARI

polrootsreal(x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1)[1] \\ Charles R Greathouse IV, Feb 11 2025