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A352594
Expansion of square root of the golden ratio phi in base phi.
1
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
OFFSET
1
COMMENTS
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]
FORMULA
sqrt(phi) = a(1) + a(2)/phi + a(3)/phi^2 + a(4)/phi^3 + a(5)/phi^4 + ...
EXAMPLE
1.001000100000010010001000001000100010101010100010101... base phi.
MATHEMATICA
RealDigits[Sqrt[GoldenRatio], GoldenRatio, 100][[1]] (* Amiram Eldar, Mar 22 2022 *)
PROG
(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))
};
print(alist(300)); \\ Chittaranjan Pardeshi, Apr 29 2022
CROSSREFS
Cf. A139339 (decimal expansion), A331692 (continued fraction), A001622 (phi).
Other numbers in base phi: A173857 (3/2), A202108 (4/sqrt(phi)).
Sequence in context: A353331 A353269 A205633 * A252488 A170956 A293449
KEYWORD
nonn,cons,base
AUTHOR
STATUS
approved