A041090 Numerators of continued fraction convergents to sqrt(53).

7, 22, 29, 51, 182, 2599, 7979, 10578, 18557, 66249, 946043, 2904378, 3850421, 6754799, 24114818, 344362251, 1057201571, 1401563822, 2458765393, 8777860001, 125348805407, 384824276222, 510173081629, 894997357851, 3195165155182, 45627309530399

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

The terms of this sequence can be constructed with the terms of sequence A086902. For the terms of the periodical sequence of the continued fraction for sqrt(53) see A010139. We observe that its period is five. The decimal expansion of sqrt(53) is A010506. - Johannes W. Meijer, Jun 12 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,364,0,0,0,0,1).

Crossrefs

Cf. A041091, A041018, A041046, A041150, A041226, A041318, A041426 and A041550.

Programs

Mathematica

Numerator[Convergents[Sqrt[53],30]] (* Harvey P. Dale, Sep 24 2013 *)
CoefficientList[Series[-(x^9 - 7 x^8 + 22 x^7 - 29 x^6 + 51 x^5 + 182 x^4 + 51 x^3 + 29 x^2 + 22 x + 7)/(x^10 + 364 x^5 - 1), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 27 2013 *)

Formula

a(5*n) = A086902(3*n+1), a(5*n+1) = (A086902(3*n+2)-A086902(3*n+1))/2, a(5*n+2) = (A086902(3*n+2)+A086902(3*n+1))/2, a(5*n+3) = A086902(3*n+2) and a(5*n+4) = A086902(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: -(x^9-7*x^8+22*x^7-29*x^6+51*x^5+182*x^4+51*x^3+29*x^2+22*x+7) / (x^10+364*x^5-1). - Colin Barker, Sep 26 2013

Extensions

More terms from Colin Barker, Sep 26 2013