login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A259597 Numerators of the other-side convergents to sqrt(7). 2
3, 5, 8, 13, 45, 82, 127, 209, 717, 1307, 2024, 3331, 11427, 20830, 32257, 53087, 182115, 331973, 514088, 846061, 2902413, 5290738, 8193151, 13483889, 46256493, 84319835, 130576328, 214896163, 737201475, 1343826622, 2081028097, 3424854719, 11748967107 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,16,0,0,0,-1).

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^7-x^6+2*x^5-3*x^4+13*x^3+8*x^2+5*x+3) / (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}]

v = Numerator[t]

LinearRecurrence[{0, 0, 0, 16, 0, 0, 0, -1}, {3, 5, 8, 13, 45, 82, 127, 209}, 40] (* Harvey P. Dale, Jan 15 2017 *)

PROG

(PARI) Vec((x^7-x^6+2*x^5-3*x^4+13*x^3+8*x^2+5*x+3)/(x^8-16*x^4+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

CROSSREFS

Cf. A041008, A041009, A259596 (denominators).

Sequence in context: A014252 A296378 A177231 * A095223 A268515 A070948

Adjacent sequences:  A259594 A259595 A259596 * A259598 A259599 A259600

KEYWORD

nonn,easy,frac

AUTHOR

Clark Kimberling, Jul 20 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 17 23:21 EDT 2019. Contains 325109 sequences. (Running on oeis4.)