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%I #130 May 08 2022 08:23:21
%S 1,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,0,
%T 0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%U 1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,0,1,0
%N Expansion of square root of the golden ratio phi in base phi.
%C What is lim_{n->oo} (1/n)*Sum_{k=1..n} a(k)? (The value is near 0.2765 at n=10^6.) - _Vaclav Kotesovec_, Mar 23 2022 [Conjecture: This value is 1/(sqrt(5)*phi) (A244847). - _Amiram Eldar_, Mar 25 2022]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Golden_ratio">Golden ratio</a>
%H Vaclav Kotesovec, <a href="/A352594/a352594.jpg">Plot of Sum_{k=1..n} a(k)/n for n = 1..2000000</a>
%H Chittaranjan Pardeshi, <a href="/A352594/a352594.py.txt">Python program</a>
%F sqrt(phi) = a(1) + a(2)/phi + a(3)/phi^2 + a(4)/phi^3 + a(5)/phi^4 + ...
%e 1.001000100000010010001000001000100010101010100010101... base phi.
%t RealDigits[Sqrt[GoldenRatio], GoldenRatio, 100][[1]] (* _Amiram Eldar_, Mar 22 2022 *)
%o (PARI)
%o alist(len) = {
%o my(phi = quadgen(5), w=phi, t =0);
%o vector(len, i,
%o w = w / phi;
%o if ( ( t + w )^2 <= phi,
%o t = t + w ;
%o 1,
%o 0))
%o };
%o print(alist(300)); \\ _Chittaranjan Pardeshi_, Apr 29 2022
%Y Cf. A139339 (decimal expansion), A331692 (continued fraction), A001622 (phi).
%Y Other numbers in base phi: A173857 (3/2), A202108 (4/sqrt(phi)).
%K nonn,cons,base
%O 1
%A _Chittaranjan Pardeshi_, Mar 21 2022
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