A180251 Decimal expansion of 6*(phi+1)/5, where phi is (1 + sqrt(5))/2.

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

Offset

1,1

Comments

This is an approximation to Pi.

6*(phi+1)/5 is not equal to Pi, although some have claimed this (see Dudley). - Kellen Myers, Oct 04 2013

References

Underwood Dudley, Mathematical Cranks, MAA, 1992, pp. 247, 292.

Alfred S. Posamentier and Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, New York, Prometheus Books, 2007, p. 119.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Hung Viet Chu, Square the Circle in One Minute, arXiv:1908.01202 [math.GM], 2019-2025.

Greg Ross, A Surprise Visitor, Futility Closet, 2011.

Index entries for algebraic numbers, degree 2.

Formula

Limit of A022089(n+2)/A022088(n) as n approaches infinity.

6*(phi + 1)/5 = 6*phi^2/5 = 3(3 + sqrt(5))/5 = 9/5 + sqrt(9/5). - Charles R Greathouse IV, Sep 13 2013

Equals 24/(5-sqrt(5))^2. - Joost Gielen, Sep 20 2013

Minimal polynomial: 25*x^2 - 90*x + 36. - Amiram Eldar, Jun 02 2026

Example

3.141640786499873817845504201238765741264371015766915434562538347246312555382...

Mathematica

RealDigits[(6/5)GoldenRatio^2, 10, 100][[1]] (* Alonso del Arte, Apr 09 2012 *)

Prog

(PARI) 3*(3+sqrt(5))/5 \\ Charles R Greathouse IV, Sep 13 2013
(Magma) (3/10)*(1 + Sqrt(5))^2; // G. C. Greubel, Jan 17 2018

Crossrefs

Cf. A022088, A022089, A001622, A000796.

Sequence in context: A069264 A347459 A064575 * A094119 A210622 A364065

Adjacent sequences: A180248 A180249 A180250 * A180252 A180253 A180254

Keyword

nonn,cons,easy

Author

Grant Garcia, Jan 16 2011

Status

approved