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A086088 Limit of ratio of consecutive terms in the tetranacci sequence A000078. 8
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

LINKS

Table of n, a(n) for n=1..102.

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'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

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.

Sequence in context: A234371 A172423 A104696 * A231986 A203126 A111506

Adjacent sequences:  A086085 A086086 A086087 * A086089 A086090 A086091

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Jul 08 2003

STATUS

approved

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Last modified October 1 10:32 EDT 2014. Contains 247508 sequences.