A010139 -id:A010139 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

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 *)

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

More terms from Colin Barker, Sep 26 2013

A041091 Denominators of continued fraction convergents to sqrt(53).

Original entry on oeis.org

1, 3, 4, 7, 25, 357, 1096, 1453, 2549, 9100, 129949, 398947, 528896, 927843, 3312425, 47301793, 145217804, 192519597, 337737401, 1205731800, 17217982601, 52859679603, 70077662204, 122937341807, 438889687625, 6267392968557, 19241068593296, 25508461561853

Offset: 0

N. J. A. Sloane

The terms of this sequence can be constructed with the terms of sequence A054413. For the terms of the periodic 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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,364,0,0,0,0,1).

Cf. A010506, A041090.

Cf. A041019, A041047, A041151, A041227, A041319, A041427 and A041551. - Johannes W. Meijer, Jun 12 2010

convert(sqrt(53), confrac, 30, cvgts): denom(cvgts); # Wesley Ivan Hurt, Dec 17 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[53], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[53], 30]] (* Vincenzo Librandi, Oct 24 2013 *)
LinearRecurrence[{0,0,0,0,364,0,0,0,0,1},{1,3,4,7,25,357,1096,1453,2549,9100},30] (* Harvey P. Dale, Nov 13 2019 *)

a(5*n) = A054413(3*n), a(5*n+1) = (A054413(3*n+1) - A054413(3*n))/2, a(5*n+2)= (A054413(3*n+1) + A054413(3*n))/2, a(5*n+3) = A054413(3*n+1) and a(5*n+4) = A054413(3*n+2)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: -(x^8-3*x^7+4*x^6-7*x^5+25*x^4+7*x^3+4*x^2+3*x+1) / (x^10+364*x^5-1). - Colin Barker, Sep 26 2013

A010506 Decimal expansion of square root of 53.

Original entry on oeis.org

7, 2, 8, 0, 1, 0, 9, 8, 8, 9, 2, 8, 0, 5, 1, 8, 2, 7, 1, 0, 9, 7, 3, 0, 2, 4, 9, 1, 5, 2, 7, 0, 3, 2, 7, 9, 3, 7, 7, 7, 6, 6, 9, 6, 8, 2, 5, 7, 6, 4, 7, 7, 4, 3, 8, 3, 7, 8, 1, 8, 1, 7, 9, 1, 2, 8, 4, 1, 1, 5, 7, 3, 8, 6, 2, 4, 9, 0, 5, 1, 8, 3, 3, 2, 9, 5, 7, 9, 4, 0, 9, 0, 8, 0, 9, 2, 6, 7, 5

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 7 followed by {3, 1, 1, 3, 14} repeated. - Harry J. Smith, Jun 06 2009

			7.280109889280518271097302491527032793777669682576477438378181791284115....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Cf. A010139 (continued fraction).

RealDigits[N[Sqrt[53],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

default(realprecision, 20080); x=sqrt(53); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010506.txt", n, " ", d));  \\ Harry J. Smith, Jun 06 2009

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

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A041091 Denominators of continued fraction convergents to sqrt(53).

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A010506 Decimal expansion of square root of 53.

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