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A259593 Numerators of the other-side convergents to sqrt(3). 2
2, 3, 7, 12, 26, 45, 97, 168, 362, 627, 1351, 2340, 5042, 8733, 18817, 32592, 70226, 121635, 262087, 453948, 978122, 1694157, 3650401, 6322680, 13623482, 23596563, 50843527, 88063572, 189750626, 328657725, 708158977, 1226567328, 2642885282, 4577611587 (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. 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 (0,4,0,-1).

FORMULA

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).

a(n) = 4*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015

G.f.: -(x^2-3*x-2) / (x^4-4*x^2+1). - Colin Barker, Jul 21 2015

EXAMPLE

For r = sqrt(3), the first 7 other-side convergents are 4, 25/8, 355/113, 688/219, 104348/33215, 208341/66317, 312689/99532. 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      1/1  < sqrt(3) <     2/1               -1

1      2/1  > sqrt(3) >     3/2                1

2      5/3  < sqrt(3) <     7/4               -1

3      7/4  > sqrt(3) >    12/7                1

4     19/11 < sqrt(3) <    26/15              -1

5     26/15 > sqrt(3) >    45/26               1

MATHEMATICA

r = Sqrt[3]; 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]

PROG

(PARI) Vec(-(x^2-3*x-2)/(x^4-4*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

CROSSREFS

Cf. A002530, A002531, A259592 (denominators).

Sequence in context: A321838 A298897 A054272 * A129016 A099163 A000676

Adjacent sequences:  A259590 A259591 A259592 * A259594 A259595 A259596

KEYWORD

nonn,easy,frac

AUTHOR

Clark Kimberling, Jul 20 2015

STATUS

approved

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Last modified August 20 18:56 EDT 2019. Contains 326154 sequences. (Running on oeis4.)