A010217 -id:A010217 - cp's OEIS frontend

A041318 Numerators of continued fraction convergents to sqrt(173).

13, 79, 92, 171, 1118, 29239, 176552, 205791, 382343, 2499849, 65378417, 394770351, 460148768, 854919119, 5589663482, 146186169651, 882706681388, 1028892851039, 1911599532427, 12498490045601, 326872340718053, 1973732534353919, 2300604875071972

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, 0, 0, 0, 2236, 0, 0, 0, 0, 1).

Cf. A041319, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550.

Cf. A010217 (continued fraction).

Numerator[Convergents[Sqrt[173], 30]] (* Vincenzo Librandi, Nov 01 2013 *)
LinearRecurrence[{0,0,0,0,2236,0,0,0,0,1},{13,79,92,171,1118,29239,176552,205791,382343,2499849},30] (* Harvey P. Dale, Jul 28 2018 *)

a(5*n) = A088316(3*n+1), a(5*n+1) = (A088316(3*n+2) - A088316(3*n+1))/2, a(5*n+2) = (A088316(3*n+2)+A088316(3*n+1))/2, a(5*n+3) = A088316(3*n+2) and a(5*n+4) = A088316(3*n+3)/2. [Johannes W. Meijer, Jun 12 2010]

G.f.: -(x^9-13*x^8+79*x^7-92*x^6+171*x^5+1118*x^4+171*x^3+92*x^2+79*x+13) / (x^10+2236*x^5-1). - Colin Barker, Nov 08 2013

More terms from Colin Barker, Nov 08 2013

A041319 Denominators of continued fraction convergents to sqrt(173).

Original entry on oeis.org

1, 6, 7, 13, 85, 2223, 13423, 15646, 29069, 190060, 4970629, 30013834, 34984463, 64998297, 424974245, 11114328667, 67110946247, 78225274914, 145336221161, 950242601880, 24851643870041, 150060105822126, 174911749692167, 324971855514293, 2124742882777925

Offset: 0

N. J. A. Sloane

The a(n) terms of this sequence can be constructed with the terms of sequence A140455. For the terms of the periodical sequence of the continued fraction for sqrt(173) see A010217. We observe that its period is five. - Johannes W. Meijer, Jun 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 2236, 0, 0, 0, 0, 1).

Cf. A041318, A010217, A041019, A041047, A041091, A041151, A041227, A041319, A041427 and A041551.

I:=[1,6,7,13,85,2223,13423,15646,29069,190060]; [n le 10 select I[n] else 2236*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 15 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[173], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[173], 30]] (* Vincenzo Librandi, Dec 15 2013 *)
LinearRecurrence[{0,0,0,0,2236,0,0,0,0,1},{1,6,7,13,85,2223,13423,15646,29069,190060},30] (* Harvey P. Dale, Sep 19 2020 *)

a(5*n) = A140455(3*n+1), a(5*n+1) = (A140455(3*n+2) - A140455(3*n+1))/2, a(5*n+2) = (A140455(3*n+2)+A140455(3*n+1))/2, a(5*n+3) = A140455(3*n+2) and a(5*n+4) = A140455(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010
G.f.: -(x^8-6*x^7+7*x^6-13*x^5+85*x^4+13*x^3+7*x^2+6*x+1) / (x^10+2236*x^5-1). - Colin Barker, Nov 12 2013
a(n) = 2236*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 15 2013

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A041318 Numerators of continued fraction convergents to sqrt(173).

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