A205769 Given an equilateral triangle T, partition each side (with the same orientation) into segments exhibiting the Golden Ratio. Let t be the resulting internal equilateral triangle t. Sequence gives decimal expansion of ratio of areas T/t.

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

Offset

1,1

Comments

A quadratic number with denominator 2 and minimal polynomial 4x^2 - 14x + 1. - Charles R Greathouse IV, Apr 21 2016

References

Alfred S. Posamentier & Ingmar Lehmann, Phi, The Glorious Golden Ratio, Prometheus Books, 2011.

Links

Table of n, a(n) for n=1..105.

Index entries for algebraic numbers, degree 2

Index entries for algebraic numbers, degree 2

Formula

= phi^2/(1 + 1/phi^2 - 1/phi).

Also, = (phi^4)/2 = 1+3*phi/2 [Clark Kimberling, Oct 24 2012]

Example

3.427050983124842272306880251548457176580463769708644293203172934...

Mathematica

x = GoldenRatio; RealDigits[x^4/(1 - x + x^2), 10, 111][[1]]

Prog

(PARI) (1+sqrt(5))^4/32 \\ Charles R Greathouse IV, Dec 12 2013

Crossrefs

Cf. A001622.

Sequence in context: A344968 A324340 A046692 * A166108 A255768 A216221

Adjacent sequences: A205766 A205767 A205768 * A205770 A205771 A205772

Keyword

nonn,cons

Author

Robert G. Wilson v, Jan 31 2012

Status

approved