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A317908
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Number of decimal places to which the n-th convergent of the continued fraction expansion of Khintchine's constant matches the correct value.
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2
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0, -1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 8, 8, 9, 11, 13, 12, 14, 15, 16, 16, 16, 18, 21, 21, 23, 24, 24, 25, 25, 26, 27, 28, 29, 30, 30, 32, 32, 33, 33, 36, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 44, 46, 47, 48, 48, 49, 50, 51, 54, 55, 56, 56, 58, 58, 60
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OFFSET
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1,4
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COMMENTS
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Decimal expansion of Khintchine's constant in A002210.
For the similar case of the number of correct decimal digits of Pi see A084407.
For the similar case of the number of correct decimal digits of log(2) see A317558.
For the number of correct binary places see A317907.
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LINKS
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FORMULA
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EXAMPLE
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n convergent decimal expansion a(n)
== ============= ==================== ====
1 2 / 1 2.0 0
2 3 / 1 3.0 -1
3 8 / 3 2.66... 1
4 43 / 16 2.687... 2
5 51 / 19 2.684... 2
6 94 / 35 2.6857... 3
7 239 / 89 2.6853... 3
8 333 / 124 2.68548... 4
9 572 / 213 2.68544... 4
10 2049 / 763 2.6854521... 6
oo lim = A002210 2.685452001065306... --
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PROG
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(Python)
i, cf = 0, []
while i <= 20100:
....cf, i = cf+[c], i+1
p0, p1, q0, q1, i, base = cf[0], 1, 1, 0, 1, 10
while i <= 20100:
....p0, p1, q0, q1, i = cf[i]*p0+p1, p0, cf[i]*q0+q1, q0, i+1
a0 = p0//q0
p0 = p0-a0*q0
i, p0, dd = 0, p0*base, [a0]
while i < 21000:
....d, p0, i = p0//q0, (p0%q0)*base, i+1
....dd = dd+[d]
n, pn0, pn1, qn0, qn1 = 1, a0, 1, 1, 0
while n <= 20000:
....p, q = pn0, qn0
....if p//q != a0:
........print(n, "- manual!")
....else:
........i, p, di = 0, (p%q)*base, a0
........while di == dd[i]:
............i, di, p = i+1, p//q, (p%q)*base
........print(n, i-1)
....n, pn0, pn1, qn0, qn1 = n+1, cf[n]*pn0+pn1, pn0, cf[n]*qn0+qn1, qn0
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CROSSREFS
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KEYWORD
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sign,base
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AUTHOR
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STATUS
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approved
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