A086088 Decimal expansion of the limit of the ratio of consecutive terms in the tetranacci sequence A000078.

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

Offset: 1

Author

Eric W. Weisstein, Jul 08 2003

Keywords

• nonn
• cons

Comments

The tetranacci constant corresponds to the Golden Section in a quadripartite division 1 = u_1 + u_2 + u_3 + u_4 of a unit line segment, i.e., if 1/u_1 = u_1/u_2 = u_2/u_3 = u_3/u_4 = c, c is the tetranacci constant. - Seppo Mustonen, Apr 19 2005

The other 3 polynomial roots of 1+x+x^2+x^3-x^4 are -0.77480411321543385... and the complex-conjugated pair -0.07637893113374572508475 +- i * 0.814703647170386526841... - R. J. Mathar, Oct 25 2008

The continued fraction expansion starts 1, 1, 12, 1, 4, 7, 1, 21, 1, 2, 1, 4, 6, 1, 10, 1, 2, 2, 1, 7, 1, 1,... - R. J. Mathar, Mar 09 2012

For n>=4, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - Vladimir Shevelev, Mar 21 2014

Note that we have: c + c^(-4) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - Bernard Schott, May 09 2022

Examples

			1.927561975...
		

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

• Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
• O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41.
• Gültekin, İnci; Deveci, Ömür, On the arrowhead-Fibonacci numbers. Open Math. 14, 1104-1113 (2016).
• 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, Tetranacci Number
• Eric Weisstein's World of Mathematics, Disk Covering Problem
• Eric Weisstein's World of Mathematics, Tetranacci Constant
• Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A000078.

k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), this sequence (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).

Programs

Mathematica

RealDigits[Root[ -1-#1-#1^2-#1^3+#1^4&, 2], 10, 110][[1]]

PARI

real(polroots(1+x+x^2+x^3-x^4)[2]) \\ Charles R Greathouse IV, Jul 19 2012

PARI

polrootsreal(1+x+x^2+x^3-x^4)[2] \\ Charles R Greathouse IV, Apr 14 2014

Formula

Equals 1/4 + sqrt(11/48 - s/72 + 7/s) + sqrt(11/24 + s/72 - 7/s + 1 / sqrt(704/507 - 128 * s/1521 + 7168 / (169 * s))) where s = (sqrt(177304464) + 7020)^(1/3). - Michal Paulovic, Oct 08 2022