A041448 Numerators of continued fraction convergents to sqrt(240).

15, 31, 945, 1921, 58575, 119071, 3630705, 7380481, 225045135, 457470751, 13949167665, 28355806081, 864623350095, 1757602506271, 53592698538225, 108942999582721, 3321882686019855, 6752708371622431, 205903133834692785, 418558976041008001, 12762672415064932815

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (0,62,0,-1).

Crossrefs

Cf. A041449, A040224.

Programs

Mathematica

Numerator[Convergents[Sqrt[240], 30]] (* or *) CoefficientList[Series[(15 + 31 x + 945 x^2 + 1921 x^3 + 945 x^4 - 31 x^5 + 15 x^6 - x^7)/(1 - 3842 x^4 + x^8), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 02 2013 *)
a0[n_] := (-15-4*Sqrt[15]+(-15+4*Sqrt[15])*(31+8*Sqrt[15])^(2*n))/(2*(31+8*Sqrt[15])^n) // Simplify
a1[n_] := (1+(31+8*Sqrt[15])^(2*n))/(2*(31+8*Sqrt[15])^n) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]}&,Range[10]]] (* Gerry Martens, Jul 10 2015 *)

Formula

G.f.: -(x+1)*(x^2-16*x-15) / ((x^2-8*x+1)*(x^2+8*x+1)). - Vincenzo Librandi, Nov 02 2013, simplified by Colin Barker, Dec 28 2013

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)]:

a0(n) = ((-15-4*sqrt(15))/(31+8*sqrt(15))^n+(-15+4*sqrt(15))*(31+8*sqrt(15))^n)/2.

a1(n) = (1/(31+8*sqrt(15))^n+(31+8*sqrt(15))^n)/2. (End)

Extensions

More terms from Colin Barker, Dec 28 2013