A259596 Denominators of the other-side convergents to sqrt(7).

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

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

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+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

Adjacent sequences: A259593 A259594 A259595 * A259597 A259598 A259599

Keyword

nonn,easy,frac

Author

Clark Kimberling, Jul 20 2015

Status

approved