A010473 -id:A010473 - cp's OEIS frontend

A248245 Egyptian fraction representation of sqrt(17) (A010473) using a greedy function.

4, 9, 84, 11142, 474347339, 1448582974451426406, 2526762018809024624337804813995389534, 28249016389028465904997590221278194109894254535234000317524709009386354668

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[ 17]]

A040012 Continued fraction for sqrt(17).

Original entry on oeis.org

4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

N. J. A. Sloane

Decimal expansion of 22/45. - Elmo R. Oliveira, Feb 06 2024

			4.123105625617660549821409855... = 4 + 1/(8 + 1/(8 + 1/(8 + 1/(8 + ...)))). - _Harry J. Smith_, Jun 03 2009

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Pages 275-276.

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

Cf. A041024/A041025 (convergents), A010473 (decimal expansion), A248245 (Egyptian fraction).

Cf. A040000.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[17],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},100,8] (* Harvey P. Dale, Jun 22 2015 *)

{ allocatemem(932245000); default(realprecision, 37000); x=contfrac(sqrt(17)); for (n=0, 20000, write("b040012.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

a(n) = 4*A040000(n). - Stefano Spezia, May 14 2023
From Elmo R. Oliveira, Feb 06 2024: (Start)
a(n) = 8 for n >= 1.
G.f.: 4*(1+x)/(1-x).
E.g.f.: 8*exp(x) - 4. (End)

A176458 Decimal expansion of 4+sqrt(17).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of 4+sqrt(17) is A010731.

This is the shape of an 8-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011

			4+sqrt(17) = 8.12310562561766054982...

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

Cf. A010473 (decimal expansion of sqrt(17)), A010731 (all 8's sequence).

Cf. A049310.

r=8; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]

4+sqrt(17) \\ Charles R Greathouse IV, Jul 24 2013

a(n) = A010473(n) for n > 1.
Equals exp(arcsinh(4)), since arcsinh(x)=log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals lim_{n->infinity} S(n, 2*sqrt(17))/S(n-1, 2*sqrt(17)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A176110 Decimal expansion of sqrt(255).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 10 2010

Continued fraction expansion of sqrt(255) is A040239.

			sqrt(255) = 15.96871942267131199907...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000
Index entries for algebraic numbers, degree 2

Cf. A010472 (decimal expansion of sqrt(15)), A010473 (decimal expansion of sqrt(17)), A040239.

RealDigits[Sqrt[255],10,120][[1]] (* Harvey P. Dale, Mar 22 2017 *)

A041024 Numerators of continued fraction convergents to sqrt(17).

Original entry on oeis.org

4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236

Offset: 0

N. J. A. Sloane

a(2*n+1) with b(2*n+1) := A041025(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 17*b^2 = +1, a(2*n) with b(2*n) := A041025(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 17*b^2 = -1 (cf. Emerson reference).

Bisection: a(2*n) = 4*S(2*n,2*sqrt(17)) = 4*A078989(n), n >= 0 and a(2*n+1) = T(n+1,33), n >= 0, with S(n,x), resp. T(n,x), Chebyshev's polynomials of the second, resp. first kind. See A049310, resp. A053120. - Wolfdieter Lang, Jan 10 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,1).

Cf. A010473, A040012, A041025.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[17],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011*)
LinearRecurrence[{8, 1}, {4, 33}, 25] (* Sture Sjöstedt, Dec 07 2011 *)
CoefficientList[Series[(4 + x)/(1 - 8 x - x^2), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

G.f.: (4+x)/(1-8*x-x^2).
a(n) = 4*A041025(n) + A041025(n-1).
a(n) = ((-i)^(n+1))*T(n+1, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.
a(n) = 8*a(n-1) + a(n-2), n > 1. - Philippe Deléham, Nov 20 2008
a(n) = ((4 + sqrt(17))^n + (4 - sqrt(17))^n)/2. - Sture Sjöstedt, Dec 08 2011

A161007 a(n+1) = 2a(n) + 16a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 20, 72, 464, 2080, 11584, 56448, 298240, 1499648, 7771136, 39536640, 203411456, 1039409152, 5333401600, 27297349632, 139929124864, 716615843840, 3672097685504, 18810048872448, 96373660712960, 493708103385088, 2529394778177536, 12958119210516480

Offset: 0

Sture Sjöstedt, Jun 02 2009

Colin Barker, Table of n, a(n) for n = 0..1000
J. Borowska, L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=2, b=4.
Index entries for linear recurrences with constant coefficients, signature (2,16).

Cf. A006131, A010473 (sqrt(17)).

[n le 2 select n-1 else 2*(Self(n-1) +8*Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 15 2022

LinearRecurrence[{2, 16}, {0, 1}, 50] (* T. D. Noe, Nov 07 2011 *)

concat(0, Vec(-x/(16*x^2+2*x-1) + O(x^40))) \\ Colin Barker, Jul 01 2015

A161007=BinaryRecurrenceSequence(2,16,0,1)
[A161007(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

a(n) = ((1+sqrt(17))^n - (1-sqrt(17))^n)/(2*sqrt(17)).
Limit_{n -> oo} a(n+1)/a(n) = 1 + sqrt(17).
G.f.: x / (1 - 2*x - 16*x^2). - Colin Barker, Jul 01 2015
a(n) = 2^(n-1)*A006131(n-1). - R. J. Mathar, Mar 08 2021
a(n) = (4*i)^n*ChebyshevU(n, -i/4). - G. C. Greubel, Oct 15 2022

A178255 Decimal expansion of (3+sqrt(17))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 24 2010

Continued fraction expansion of (3+sqrt(17))/2 is A109007.
a(n) = A082486(n) for n > 1.
The rectangle R whose shape (i.e., length/width) is (3+sqrt(17))/2 can be partitioned into rectangles of shapes 3 and 3/2 in a manner that matches the periodic continued fraction [3, 3/2, 3, 3/2, ...].  R can also be partitioned into squares so as to match the periodic continued fraction [3, 1, 1, 3, 1, 1,...].  For details, see A188635. - Clark Kimberling, May 07 2011
The positive eigenvalue of the real symmetric 2 X 2 matrix M defined by M(i,j) = max(i,j) = [(1  2), (2  2)] is (3+sqrt(17))/2, while the negative one is (3-sqrt(17))/2. For a generalization, see A085984. - Bernard Schott, Apr 13 2020
A quadratic integer with minimal polynomial x^2 - 3x - 2. - Charles R Greathouse IV, Apr 14 2020
The positive root of x^2 - 3^x - 2. The negative root is -(-3 + sqrt(17))/2 = -0.56155...  - Wolfdieter Lang, Dec 10 2022

			(3+sqrt(17))/2 = 3.56155281280883027491...

Index entries for algebraic numbers, degree 2

Cf. A082486 (decimal expansion of (5+sqrt(17))/2), A010473 (decimal expansion of sqrt(17)), A109007 (repeat 3, 1, 1), A085984.

FromContinuedFraction[{3, 3/2, {3, 3/2}}]
ContinuedFraction[%, 100] (* [3,1,1,3,1,1,...] *)
RealDigits[N[%%, 120]]    (* A178255 *)
N[%%%, 40]
(* Clark Kimberling, May 07 2011 *)

(3+sqrt(17))/2 \\ Charles R Greathouse IV, Apr 14 2020

A010504 Decimal expansion of square root of 51.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

			7.141428428542849997999399811367265278766171159902733833208430882765820...

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

Cf. A040043 (continued fraction).

Digits:=100; evalf(sqrt(51)); # Wesley Ivan Hurt, Mar 04 2014

RealDigits[N[Sqrt[51],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

{ default(realprecision, 20080); x=sqrt(51); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010504.txt", n, " ", d)); } \\ Harry J. Smith, Jun 06 2009

Equals A002194*A010473. - R. J. Mathar, Jun 08 2025

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 *)

A174930 Decimal expansion of (4+sqrt(17))/8.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 02 2010

Continued fraction expansion of (4+sqrt(17))/8 is A174927.

			(4+sqrt(17))/8 = 1.01538820320220756872...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010473 (decimal expansion of sqrt(17)), A174927 (repeat 1, 64).

cp's OEIS Frontend

A248245 Egyptian fraction representation of sqrt(17) (A010473) using a greedy function.

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A040012 Continued fraction for sqrt(17).

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A176458 Decimal expansion of 4+sqrt(17).

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A176110 Decimal expansion of sqrt(255).

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A041024 Numerators of continued fraction convergents to sqrt(17).

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A161007 a(n+1) = 2*a(n) + 16*a(n-1), a(0)=0, a(1)=1.

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A178255 Decimal expansion of (3+sqrt(17))/2.

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A010504 Decimal expansion of square root of 51.

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A161007 a(n+1) = 2a(n) + 16a(n-1), a(0)=0, a(1)=1.