OFFSET
1,2
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
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).
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)
EXAMPLE
The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7, ...
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 *)
PROG
(PARI) Vec(x*(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4) + O(x^60)) \\ Colin Barker, Jul 28 2017
(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
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Clark Kimberling, Aug 27 2008
STATUS
approved