A079934 Greedy frac multiples of sqrt(2): a(1)=1, Sum_{n>=0} frac(a(n)*x)=1 at x=sqrt(2).

1, 3, 5, 10, 17, 29, 46, 99, 169, 268, 577, 985, 1562, 3363, 5741, 9104, 19601, 33461, 53062, 114243, 195025, 309268, 665857, 1136689, 1802546, 3880899, 6625109, 10506008, 22619537, 38613965, 61233502, 131836323, 225058681, 356895004, 768398401, 1311738121

Offset: 1

Author

Benoit Cloitre and Paul D. Hanna, Jan 20 2003

Keywords

• nonn
• easy

Comments

The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

Examples

			a(4) = 10 since frac(1x) + frac(3x) + frac(5x) + frac(10x) < 1, while frac(1x) + frac(3x) + frac(5x) + frac(k*x) > 1 for all k > 5 and k < 10.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A000129 (Pell numbers), A078343, A079935, A079936.

Programs

Mathematica

CoefficientList[Series[(1 + 3*z + 5*z^2 + 4*z^3 - z^4 - z^5 - 13*z^6 + 2*z^9)/(1 - 6*z^3 + z^6), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

PARI

x='x+O('x^50); Vec(x*(2*x^9 -13*x^6 -x^5 -x^4 +4*x^3 +5*x^2 +3*x +1)/(x^6-6*x^3 +1)) \\ G. C. Greubel, Sep 22 2017

Formula

For n > 0, a(3*n) = A000129(2*n+1).

a(3*n+2) = a(3*n) + A000129(2*n+2).

a(3*n+4) = a(3*n+2) + a(3*n+3).

a(3*n) = ceiling((3+2*sqrt(2))^n*(2+sqrt(2))/4).

a(3*n+2)/a(3*n+1) -> 1/sqrt(2).

a(3*n+1)/a(3*n) -> 3-sqrt(2).

a(3*n)/a(3*n-1) -> (8+5*sqrt(2))/7.

G.f.: x*(2*x^9 - 13*x^6 - x^5 - x^4 + 4*x^3 + 5*x^2 + 3*x + 1) / (x^6 - 6*x^3 + 1). - Colin Barker, Jun 16 2013