A143642 Numerators of principal and intermediate convergents to 3^(1/2).

1, 2, 3, 5, 7, 12, 19, 26, 45, 71, 97, 168, 265, 362, 627, 989, 1351, 2340, 3691, 5042, 8733, 13775, 18817, 32592, 51409, 70226, 121635, 191861, 262087, 453948, 716035, 978122, 1694157, 2672279, 3650401, 6322680, 9973081, 13623482, 23596563, 37220045

Offset

1,2

References

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-1).

Formula

From Colin Barker, Jul 28 2017: (Start)

G.f.: x*(1 + x)*(1 + x + 2*x^2 - x^3) / (1 - 4*x^3 + x^6).

a(n) = 4*a(n-3) - a(n-6) for n>6.

(End)

Example

The first few principal and intermediate convergents to 3^(1/2) are 1/1, 2/1, 3/2, 5/3, 7/4, 12/7, ...

Mathematica

LinearRecurrence[{0, 0, 4, 0, 0, -1}, {1, 2, 3, 5, 7, 12}, 40] (* Harvey P. Dale, May 12 2018 *)

Prog

(PARI) Vec(x*(1 + x)*(1 + x + 2*x^2 - x^3) / (1 - 4*x^3 + x^6) + O(x^60)) \\ Colin Barker, Jul 28 2017

Crossrefs

Cf. A140827 (denominators).

Sequence in context: A374746 A345669 A335093 * A192685 A293543 A060986

Adjacent sequences: A143639 A143640 A143641 * A143643 A143644 A143645

Keyword

nonn,frac,easy

Author

Clark Kimberling, Aug 27 2008

Status

approved