login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A086088 Decimal expansion of the limit of the ratio of consecutive terms in the tetranacci sequence A000078. 12
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
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
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
EXAMPLE
1.927561975...
MATHEMATICA
RealDigits[Root[ -1-#1-#1^2-#1^3+#1^4&, 2], 10, 110][[1]]
PROG
(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
CROSSREFS
Cf. A000078.
k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), this sequence (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).
Sequence in context: A234371 A172423 A104696 * A231986 A347329 A203126
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 08 2003
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 21 06:00 EST 2024. Contains 370219 sequences. (Running on oeis4.)