

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
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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(n1) + aa(n2) + aa(n3))  aa(n4) 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(n1) = this constant; see for instance A192234, A192237, A317973, A317974, A317975, A317976.


LINKS

A.H.M. Smeets, Table of n, a(n) for n = 1..9999 (terms 1..3000 from Muniru A Asiru)


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



