A143607 Numerators of principal and intermediate convergents to 2^(1/2).

1, 3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521, 2273378, 3880899, 5488420, 9369319, 13250218, 22619537, 31988856, 54608393

Offset: 1

Author

Clark Kimberling, Aug 27 2008

Keywords

• nonn
• frac
• easy

Comments

Sequence is essentially A082766 (by omitting two terms A082766(0) and A082766(2)). - L. Edson Jeffery, Apr 04 2011

a(n) = A119016(n+2) for n>=2. - Georg Fischer, Oct 07 2018

Examples

			The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7, ...
		

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 (0,2,0,1).

Crossrefs

Cf. A002965 (denominators), A082766, A119016.

Programs

GAP

a:=[1,3,4,7,10];; for n in [6..40] do a[n]:=2*a[n-2]+a[n-4]; od; a; # Muniru A Asiru, Oct 07 2018

Maple

seq(coeff(series(x*(1+x)*(1+2*x+x^3)/(1-2*x^2-x^4),x,n+1), x, n), n = 1 .. 40); # Muniru A Asiru, Oct 07 2018

Mathematica

CoefficientList[Series[(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4), {x, 0, 50}], x] (* or *)
LinearRecurrence[{0, 2, 0, 1}, {1, 3, 4, 7, 10}, 40] (* Stefano Spezia, Oct 08 2018; signature amended by Georg Fischer, Apr 02 2019 *)

PARI

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

Formula

From Colin Barker, Jul 28 2017: (Start)

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

a(n) = 2*a(n-2) + a(n-4) for n>5.

(End)