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

%I A180251 #56 Feb 21 2025 19:37:00
%S A180251 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,
%T A180251 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,
%U A180251 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
%N A180251 Decimal expansion of 6*(phi+1)/5, where phi is (1 + sqrt(5))/2.
%C A180251 This is an approximation to Pi.
%C A180251 6*(phi+1)/5 is not equal to Pi, although some have claimed this (see Dudley). - _Kellen Myers_, Oct 04 2013
%D A180251 Underwood Dudley, Mathematical Cranks, MAA 1992, pp. 247, 292.
%D A180251 Alfred S. Posamentier and Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, New York, Prometheus Books, 2007, p. 119.
%H A180251 G. C. Greubel, <a href="/A180251/b180251.txt">Table of n, a(n) for n = 1..10000</a>
%H A180251 Hung Viet Chu, <a href="https://arxiv.org/abs/1908.01202">Square the Circle in One Minute</a>, arXiv:1908.01202 [math.GM], 2019.
%H A180251 Futility Closet, <a href="http://www.futilitycloset.com/2011/01/16/a-surprise-visitor/">A Surprise Visitor</a>
%F A180251 Limit of A022089(n+2)/A022088(n) as n approaches infinity.
%F A180251 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
%F A180251 Equals 24/(5-sqrt(5))^2. - _Joost Gielen_, Sep 20 2013
%e A180251 3.141640786499873817845504201238765741264371015766915434562538347246312555382...
%t A180251 RealDigits[(6/5)GoldenRatio^2, 10, 100][[1]] (* _Alonso del Arte_, Apr 09 2012 *)
%o A180251 (PARI) 3*(3+sqrt(5))/5 \\ _Charles R Greathouse IV_, Sep 13 2013
%o A180251 (Magma) (3/10)*(1 + Sqrt(5))^2; // _G. C. Greubel_, Jan 17 2018
%Y A180251 Cf. A022088, A022089, A001622, A000796.
%K A180251 nonn,cons,easy
%O A180251 1,1
%A A180251 _Grant Garcia_, Jan 16 2011