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A318605 Decimal expansion of geometric progression constant for Coxeter's Loxodromic Sequence of Tangent Circles. 9
2, 8, 9, 0, 0, 5, 3, 6, 3, 8, 2, 6, 3, 9, 6, 3, 8, 1, 2, 4, 5, 7, 0, 0, 9, 2, 9, 6, 1, 0, 3, 1, 2, 9, 6, 0, 9, 4, 3, 5, 9, 1, 7, 2, 2, 1, 6, 4, 5, 8, 5, 9, 1, 1, 0, 7, 5, 2, 0, 8, 9, 0, 0, 5, 2, 4, 4, 5, 5, 8, 0, 3, 8, 3, 5, 4, 9, 7, 0, 4, 6, 1, 5, 3, 7, 5, 9, 1, 4, 1, 9, 1, 7, 7, 8, 5, 1, 3, 9, 6, 0, 2, 3, 2, 6, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This constant and its reciprocal are the real solutions of x^4 - 2*x^3 - 2*x^2 - 2*x + 1 = (x^2 - (sqrt(5)+1)*x + 1)*(x^2 + (sqrt(5)-1)*x + 1 = 0.

This constant and its reciprocal are the solutions of x^2 - (1+sqrt(5))x + 1 = 0.

Decimal expansion of the largest x satisfying x^2 - (1+sqrt(5))x + 1 = 0.

For sequences of type aa(n) = 2*(aa(n-1) + aa(n-2) + aa(n-3)) - aa(n-4) for arbitrary initial terms (except the trivial all zero), i.e., linear recurrence relations of order 4 with signature (2,2,2,-1), lim_{n -> infinity} aa(n)/aa(n-1) = this constant; see for instance A192234, A192237, A317973, A317974, A317975, A317976.

LINKS

Muniru A Asiru, Table of n, a(n) for n = 1..3000

FORMULA

Equals A001622 + A139339, i.e., phi + sqrt(phi) where phi is the golden ratio.

EXAMPLE

2.8900536382639638124570092961031296094359...

MAPLE

evalf[180]((1+sqrt(5))/2+sqrt((1+sqrt(5))/2)); # Muniru A Asiru, Nov 21 2018

MATHEMATICA

RealDigits[GoldenRatio + Sqrt[GoldenRatio], 10 , 120][[1]] (* Amiram Eldar, Nov 22 2018 *)

PROG

(PARI) ((1+sqrt(5))/2 + sqrt((1+sqrt(5))/2)) \\ Michel Marcus, Nov 21 2018

CROSSREFS

Cf. A001622, A139339, A192234, A192237, A317973, A317974, A317975, A317976.

Sequence in context: A012649 A009627 A009677 * A021350 A016641 A155748

Adjacent sequences:  A318602 A318603 A318604 * A318606 A318607 A318608

KEYWORD

nonn,cons

AUTHOR

A.H.M. Smeets, Sep 07 2018

STATUS

approved

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Last modified August 26 02:19 EDT 2019. Contains 326324 sequences. (Running on oeis4.)