A010506 -id:A010506 - cp's OEIS frontend

A248278 Egyptian fraction representation of sqrt(53) (A010506) using a greedy function.

7, 4, 34, 1433, 3473810, 16229351336487, 949514635841230182654078450, 2889844410885034994651072554166092838631734010754362047, 90303610423494587890114446343335205731154007285533876023746429382538260256932049359769872513411427600496627202

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 53]]

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

Original entry on oeis.org

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

A176439 Decimal expansion of (7+sqrt(53))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (7+sqrt(53))/2 is A010727.
This is the shape of a 7-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011
c^n = c * A054413(n-1) + A054413(n-2), where c = (7+sqrt(53))/2. - Gary W. Adamson, Apr 14 2024

			(7+sqrt(53))/2 = 7.14005494464025913554...

Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A010506 (decimal expansion of sqrt(53)), A010727 (all 7's sequence).

Cf. A049310.

r=7; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(7+Sqrt[53])/2,10,120][[1]] (* Harvey P. Dale, Nov 03 2024 *)

(7+sqrt(53))/2 \\ Charles R Greathouse IV, Jul 24 2013

Equals lim_{n->oo} S(n, sqrt(53))/S(n-1, sqrt(53)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

Positive solution of x^2 - 7*x - 1 = 0. - Hugo Pfoertner, Apr 14 2024

A010139 Continued fraction for sqrt(53).

Original entry on oeis.org

7, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3

Offset: 0

N. J. A. Sloane

			7.280109889280518271097302491... = 7 + 1/(3 + 1/(1 + 1/(1 + 1/(3 + ...)))). - _Harry J. Smith_, Jun 06 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 1).

Cf. A010506 Decimal expansion. - Harry J. Smith, Jun 06 2009

ContinuedFraction[Sqrt[53],300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)
PadRight[{7},120,{14,3,1,1,3}] (* Harvey P. Dale, May 22 2016 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(53)); for (n=0, 20000, write("b010139.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 06 2009

A177271 Decimal expansion of sqrt(635918528029).

Original entry on oeis.org

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

Offset: 6

Klaus Brockhaus, May 07 2010

Continued fraction expansion of sqrt(635918528029) is 797445 followed by (repeat 398722, 1, 1, 398722, 1594890).

sqrt(635918528029) = sqrt(17)*sqrt(53)*sqrt(193)*sqrt(3656953).

			sqrt(635918528029) = 797445.00000250800995679558...

Daniel Starodubtsev, Table of n, a(n) for n = 6..10000

Cf. A010473 (decimal expansion of sqrt(17)), A010506 (decimal expansion of sqrt(53)), A177272 (decimal expansion of sqrt(193)), A177273 (decimal expansion of sqrt(3656953)), A177274 (continued fraction expansion of (684125+sqrt(635918528029))/1033802), A177270 (decimal expansion of (684125+sqrt(635918528029))/1033802).

First[RealDigits[Sqrt[635918528029],10,120]] (* Paolo Xausa, Jan 09 2024 *)

A378798 Decimal expansion of 16/(207*sqrt(53)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 07 2024

			0.010617241657869683013176268331458620426619516445285...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.23, p. 174.

Cf. A010506.

RealDigits[16/(207Sqrt[53]),10,100][[1]]

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A248278 Egyptian fraction representation of sqrt(53) (A010506) using a greedy function.

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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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A176439 Decimal expansion of (7+sqrt(53))/2.

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A010139 Continued fraction for sqrt(53).

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A177271 Decimal expansion of sqrt(635918528029).

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A378798 Decimal expansion of 16/(207*sqrt(53)).

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