A143609 Numerators of the upper principal and intermediate convergents to 2^(1/2).

2, 3, 10, 17, 58, 99, 338, 577, 1970, 3363, 11482, 19601, 66922, 114243, 390050, 665857, 2273378, 3880899, 13250218, 22619537, 77227930, 131836323, 450117362, 768398401, 2623476242, 4478554083, 15290740090, 26102926097, 89120964298, 152139002499

Offset: 1

Author

Clark Kimberling, Aug 27 2008

Keywords

• nonn
• easy

Comments

The upper principal and intermediate convergents to 2^(1/2), beginning with

2/1, 3/2, 10/7, 17/12, 58/41, form a strictly decreasing sequence;

essentially, numerators=A143609 and denominators=A084068.

Examples

			2*x + 3*x^2 + 10*x^3 + 17*x^4 + 58*x^5 + 99*x^6 + 338*x^7 + 577*x^8 + ...
		

References

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

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Creighton Kenneth Dement, Comments on A143608 and A143609
• 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 (0,6,0,-1).

Crossrefs

Cf. A001541, A001653, A010914, A084068.

Programs

Mathematica

Rest@ CoefficientList[Series[x (2 + 3 x - 2 x^2 - x^3)/(1 - 6 x^2 + x^4), {x, 0, 30}], x] (* Michael De Vlieger, Mar 27 2016 *)

PARI

{a(n) = if( n<1, 0, polcoeff( x * (2 + 3*x - 2*x^2 - x^3) / (1 - 6*x^2 + x^4) + x * O(x^n), n))} /* Michael Somos, Sep 03 2013 */

PARI

x='x+O('x^99); Vec(x*(2+3*x-2*x^2-x^3)/(1-6*x^2+x^4)) \\ Altug Alkan, Mar 27 2016

Formula

a(n) = 6 * a(n-2) - a(n-4). a(2*n) = A001541(n) if n>0. a(2*n + 1) = 2 * A001653(n + 1).- Michael Somos, Sep 03 2013

G.f.: x * (2 + 3*x - 2*x^2 - x^3) / (1 - 6*x^2 + x^4). - Michael Somos, Sep 03 2013

a(n) = (2+sqrt(2)+(-1)^n*(-2+sqrt(2)))*((-1+sqrt(2))^n+(1+sqrt(2))^n)/(4*sqrt(2)). - Colin Barker, Mar 27 2016