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

Author

Grant Garcia, Jan 16 2011

Keywords

• nonn
• cons
• easy

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

Examples

			3.141640786499873817845504201238765741264371015766915434562538347246312555382...
		

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.
• Futility Closet, A Surprise Visitor

Crossrefs

Cf. A022088, A022089, A001622, A000796.

Programs

Magma

(3/10)*(1 + Sqrt(5))^2; // G. C. Greubel, Jan 17 2018

Mathematica

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

PARI

3*(3+sqrt(5))/5 \\ Charles R Greathouse IV, Sep 13 2013

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