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A259596 Denominators of the other-side convergents to sqrt(7). 2
1, 2, 3, 5, 17, 31, 48, 79, 271, 494, 765, 1259, 4319, 7873, 12192, 20065, 68833, 125474, 194307, 319781, 1097009, 1999711, 3096720, 5096431, 17483311, 31869902, 49353213, 81223115, 278635967, 507918721, 786554688, 1294473409, 4440692161, 8094829634 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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. The 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.
LINKS
FORMULA
p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).
a(n) = 16*a(n-4) - a(n-8) for n>7. - Colin Barker, Jul 21 2015
G.f.: -(x+1)*(x^2-x-1)*(x^4+3*x^2+1) / (x^8-16*x^4+1). - Colin Barker, Jul 21 2015
EXAMPLE
For r = sqrt(7), 3, 5/2, 8/3, 13/5, 45/17, 82/31, 127/48. 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 < sqrt(7) < 3/1 -1
1 3/1 > sqrt(7) > 5/2 1
2 5/2 < sqrt(7) < 8/3 -1
3 8/3 > sqrt(7) > 13/5 1
4 37/14 < sqrt(7) < 45/17 -1
5 45/17 > sqrt(7) > 83/31 1
MATHEMATICA
r = Sqrt[7]; 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]
LinearRecurrence[{0, 0, 0, 16, 0, 0, 0, -1}, {1, 2, 3, 5, 17, 31, 48, 79}, 40] (* Harvey P. Dale, Jun 03 2017 *)
PROG
(PARI) Vec(-(x+1)*(x^2-x-1)*(x^4+3*x^2+1)/(x^8-16*x^4+1) + O(x^50)) \\ Colin Barker, Jul 21 2015
CROSSREFS
Cf. A041008, A041009, A259597 (numerators).
Sequence in context: A029972 A077498 A118958 * A189536 A163588 A270539
KEYWORD
nonn,easy,frac
AUTHOR
Clark Kimberling, Jul 20 2015
STATUS
approved

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Last modified April 14 09:06 EDT 2024. Contains 371657 sequences. (Running on oeis4.)