A041018 Numerators of continued fraction convergents to sqrt(13).

3, 4, 7, 11, 18, 119, 137, 256, 393, 649, 4287, 4936, 9223, 14159, 23382, 154451, 177833, 332284, 510117, 842401, 5564523, 6406924, 11971447, 18378371, 30349818, 200477279, 230827097, 431304376, 662131473

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,36,0,0,0,0,1).

Formula

From Johannes W. Meijer, Jun 12 2010: (Start)

a(5*n) = A006497(3*n+1),

a(5*n+1) = (A006497(3*n+2)-A006497(3*n+1))/2,

a(5*n+2) = (A006497(3*n+2)+A006497(3*n+1))/2,

a(5*n+3) = A006497(3*n+2),

a(5*n+4) = A006497(3*n+3)/2.

(End)

G.f.: (3 + 4*x + 7*x^2 + 11*x^3 + 18*x^4 + 11*x^5 - 7*x^6 + 4*x^7 - 3*x^8 + x^9)/(1 - 36*x^5 - x^10). - Peter J. C. Moses, Jul 29 2013

a(n) = A010122(n)*a(n-1)+a(n-2) with a(0)=3, a(-1)=1. - Paul Weisenhorn, Aug 19 2018

Maple

a[0]:=3: a[-1]:=1: b(0):=6: b(1):=1; b(2):=1: b(3):=1: b(4):=1:
for n from 1 to 100 do  k:=n mod 5:
   a[n]:=b(k)*a[n-1]+a[n-2]:
   printf("%12d", a[n]):
end do: # Paul Weisenhorn, Aug 17 2018

Mathematica

Numerator[Convergents[Sqrt[13], 30]] (* Vincenzo Librandi, Oct 27 2013 *)
CoefficientList[Series[(3 + 4*x + 7*x^2 + 11*x^3 + 18*x^4 + 11*x^5 - 7*x^6 + 4*x^7 - 3*x^8 + x^9)/(1 - 36*x^5 - x^10), {x, 0, 50}], x] (* Stefano Spezia, Aug 31 2018 *)

Crossrefs

Cf. A010122 (continued fraction for sqrt(13)).

Cf. A010470, A041019 (denominators), A041046, A041090, A041150, A041226, A041318, A041426 and A041550.

Sequence in context: A293420 A041739 A042593 * A361907 A072255 A049863

Adjacent sequences: A041015 A041016 A041017 * A041019 A041020 A041021

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved