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A352594
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Expansion of square root of the golden ratio phi in base phi.
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1
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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, 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, 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
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OFFSET
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1
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COMMENTS
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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]
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LINKS
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FORMULA
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sqrt(phi) = a(1) + a(2)/phi + a(3)/phi^2 + a(4)/phi^3 + a(5)/phi^4 + ...
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EXAMPLE
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1.001000100000010010001000001000100010101010100010101... base phi.
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MATHEMATICA
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RealDigits[Sqrt[GoldenRatio], GoldenRatio, 100][[1]] (* Amiram Eldar, Mar 22 2022 *)
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PROG
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(PARI)
alist(len) = {
my(phi = quadgen(5), w=phi, t =0);
vector(len, i,
w = w / phi;
if ( ( t + w )^2 <= phi,
t = t + w ;
1,
0))
};
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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