Sturmian word: Difference between revisions
julia example
m (it's the inverse golder ratio) |
(julia example) |
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In summary, <math>floor(k\sqrt a) = floor(mk/n)</math> where <math>m/n</math> is the first continued fraction approximant to <math>\sqrt a</math> with a denominator <math>n \geq k</math>
=={{header|Julia}}==
{{trans|Phix}}
<syntaxhighlight lang="julia">function sturmian_word(m, n)
m > n && return replace(sturmian_word(n, m), '0' => '1', '1' => '0')
res = ""
k, prev = 1, 0
while rem(k * m, n) > 0
curr = (k * m) ÷ n
res *= prev == curr ? "0" : "10"
prev = curr
k += 1
end
return res
end
function fibWord(n)
Sn_1, Sn, tmp = "0", "01", ""
for _ in 2:n
tmp = Sn
Sn *= Sn_1
Sn_1 = tmp
end
return Sn
end
const fib = fibWord(7)
const sturmian = sturmian_word(13, 21)
@assert fib[begin:length(sturmian)] == sturmian
println(" $sturmian <== 13/21")
""" return the kth convergent """
function cfck(a, b, m, n, k)
p = [0, 1]
q = [1, 0]
r = (sqrt(a) * b + m) / n
for _ in 1:k
whole = Int(trunc(r))
pn = whole * p[end] + p[end-1]
qn = whole * q[end] + q[end-1]
push!(p, pn)
push!(q, qn)
r = 1/(r - whole)
end
return [p[end], q[end]]
end
println(" $(sturmian_word(cfck(5, 1, -1, 2, 8)...)) <== 1/phi (8th convergent golden ratio)")
</syntaxhighlight>{{out}}
<pre>
01001010010010100101001001010010 <== 13/21
01001010010010100101001001010010 <== 1/phi (8th convergent golden ratio)
</pre>
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