A040224 -id:A040224 - cp's OEIS frontend

A041449 Denominators of continued fraction convergents to sqrt(240).

1, 2, 61, 124, 3781, 7686, 234361, 476408, 14526601, 29529610, 900414901, 1830359412, 55811197261, 113452753934, 3459393815281, 7032240384496, 214426605350161, 435885451084818, 13290990137894701, 27017865726874220, 823826961944121301, 1674671789615116822

Offset: 0

N. J. A. Sloane

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

Cf. A041448, A040224.

I:=[1,2,61,124]; [n le 4 select I[n] else 62*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 18 2013

Denominator[Convergents[Sqrt[240], 30]] (* Vincenzo Librandi, Dec 18 2013 *)
a0[n_] := Sqrt[2+(31-8*Sqrt[15])^(1+2*n)+(31+8*Sqrt[15])^(1+2*n)]/8 // Simplify
a1[n_] := 2*Sum[a0[i], {i, 0, n}]
Flatten[MapIndexed[{a0[#-1],a1[#-1]}&,Range[11]]] (* Gerry Martens, Jul 10 2015 *)

G.f.: -(x^2-2*x-1) / ((x^2-8*x+1)*(x^2+8*x+1)). - Colin Barker, Nov 17 2013
a(n) = 62*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Dec 18 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n-1),a1(n-1)] for n>0:
a0(n) = sqrt(2+(31-8*sqrt(15))^(2*n+1)+(31+8*sqrt(15))^(2*n+1))/8.
a1(n) = 2*sum(i=0,n,a0(i)). (End)

More terms from Colin Barker, Nov 17 2013

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

Cf. A041449, A040224.

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 *)

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)

More terms from Colin Barker, Dec 28 2013

cp's OEIS Frontend

A041449 Denominators of continued fraction convergents to sqrt(240).

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