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A317908 Number of decimal places to which the n-th convergent of the continued fraction expansion of Khintchine's constant matches the correct value. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
FORMULA
Limit_{n -> oo} (a(n)/n) = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995.
EXAMPLE
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... --
PROG
(Python)
i, cf = 0, []
while i <= 20100:
....c = A002211(i)
....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
CROSSREFS
Sequence in context: A198318 A100881 A053263 * A056039 A181972 A356737
KEYWORD
sign,base
AUTHOR
A.H.M. Smeets, Aug 10 2018
STATUS
approved

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Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)