Revision History for A259589
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strikethrough text is a deletion.)
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#6 by N. J. A. Sloane at Sun Jul 19 11:04:27 EDT 2015
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#5 by Jon E. Schoenfield at Fri Jul 17 16:02:51 EDT 2015
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#4 by Jon E. Schoenfield at Fri Jul 17 16:02:48 EDT 2015
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| COMMENTS
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Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. ), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. ]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to
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| EXAMPLE
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For r = e, the first 13 other-side convergents are 3/1, 5/2, 11/4, 19/7, 30/11, 106/39, 193/71, 299/110, 1457/536, 2721/1001, 4178/1537, 25946/9545, 49171/18089.
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| STATUS
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proposed
editing
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#3 by Clark Kimberling at Fri Jul 17 15:53:20 EDT 2015
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#2 by Clark Kimberling at Fri Jul 17 15:32:03 EDT 2015
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| NAME
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allocatedNumerators of the other-side forconvergents Clarkto Kimberlinge.
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| DATA
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3, 5, 11, 19, 30, 106, 193, 299, 1457, 2721, 4178, 25946, 49171, 75117, 566827, 1084483, 1651310, 14665106, 28245729, 42910835, 438351041, 848456353, 1286807394, 14862109042, 28875761731, 43737870773, 563501581931, 1098127402131, 1661628984062
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| OFFSET
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0,1
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| COMMENTS
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Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to
r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0.
Closeness of P(i)/Q(i) to r is indicated by |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.
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| FORMULA
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p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
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| EXAMPLE
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For r = e, the first 13 other-side convergents are 3/1, 5/2, 11/4, 19/7, 30/11, 106/39, 193/71, 299/110, 1457/536, 2721/1001, 4178/1537, 25946/9545, 49171/18089.
A comparison of convergents with other-side convergents:
i p(i)/q(i) P(i)/Q(i) p(i)*Q(i) - P(i)*q(i)
0 2/1 < e < 3/1 -1
1 3/1 > e > 5/2 1
2 8/3 < e < 11/4 -1
3 11/4 > e > 19/7 1
4 19/7 < e < 30/11 -1
5 87/32 > e > 106/39 1
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| MATHEMATICA
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r = E; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
u = Denominator[t] (* A259588 *)
v = Numerator[t] (* A259589 *)
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| CROSSREFS
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Cf. A259588, A007676, A007677.
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| KEYWORD
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allocated
nonn,easy,frac
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| AUTHOR
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Clark Kimberling, Jul 17 2015
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| STATUS
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approved
editing
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#1 by Clark Kimberling at Tue Jun 30 15:27:11 EDT 2015
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| NAME
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allocated for Clark Kimberling
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| KEYWORD
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allocated
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| STATUS
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approved
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