A010467 -id:A010467 - cp's OEIS frontend

A248239 Egyptian fraction representation of sqrt(10) (A010467) using a greedy function.

3, 7, 52, 5271, 32510519, 1551821465402536, 2553352811042166137014681056617, 6785214292790116540717856342564735260380655042140053309985580, 57499324177051573068556985649019772314982410954417460069917198506894068347777607349711324456505504280305966462257432295349

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[ 10]]

A005667 Numerators of continued fraction convergents to sqrt(10).

Original entry on oeis.org

1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, 243289797, 1499219281, 9238605483, 56930852179, 350823718557, 2161873163521, 13322062699683, 82094249361619, 505887558869397, 3117419602578001, 19210405174337403, 118379850648602419

Offset: 0

N. J. A. Sloane, R. K. Guy

a(2*n+1) with b(2*n+1) := A005668(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = -1, a(2*n) with b(2*n) := A005668(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = +1 (cf. Emerson reference).
Bisection: a(2*n) = T(n,19) = A078986(n), n >= 0 and a(2*n+1) = 3*S(2*n, 2*sqrt(10)), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. See A053120, resp. A049310.
The initial 1 corresponds to a denominator 0 in A005668. But according to standard conventions, a continued fraction starts with b(0) = integer part of the number, and the sequence of convergents p(n)/q(n) start with (p(0),q(0)) = (b(0),1). A fraction 1/0 has no mathematical meaning, the only justification is that initial terms p(-1) = 1, q(-1) = 0 are consistent with the recurrent relations p(n) = b(n)*p(n-1) + b(n-2) and the same for q(n). - M. F. Hasler, Nov 02 2019

			G.f. = 1 + 3*x + 19*x^2 + 117*x^3 + 721*x^4 + 4443*x^5 + 27379*x^6 + ... - _Michael Somos_, Jul 14 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Thm. 1, p. 233.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Tian-Xiao He and Peter J.-S. Shiue, Identities for linear recursive sequences of order 2, Elect. Res. Archive (2021) Vol. 29, No. 5, 3489-3507.
Tanya Khovanova, Recursive Sequences
Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,1).

Cf. A010467, A040006, A084134, A005668 (denominators).

I:=[1, 3]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 09 2013

A005667:=(-1+3*z)/(-1+6*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

Join[{1},Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[10],n]]],{n,1,30}]] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
CoefficientList[Series[(1-3x)/(1-6x-x^2), {x,0,30}], x] (* Vincenzo Librandi, Jun 09 2013 *)
Join[{1},Numerator[Convergents[Sqrt[10],30]]] (* or *) LinearRecurrence[ {6,1},{1,3},30] (* Harvey P. Dale, Aug 22 2016 *)
a[ n_] := (-I)^n ChebyshevT[ n, 3 I]; (* Michael Somos, Jul 14 2018 *)
LucasL[Range[0,30], 6]/2 (* G. C. Greubel, Jun 06 2019 *)

a(n)=([0,1;1,6]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

((1-3*x)/(1-6*x-x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019

a(n) = 6*a(n-1) + a(n-2).
G.f.: (1-3*x)/(1-6*x-x^2).
a(n) = ((-i)^n)*T(n, 3*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1.
From Paul Barry, Nov 15 2003: (Start)
Binomial transform of A084132.
E.g.f.: exp(3*x)*cosh(sqrt(10)*x).
a(n) = ((3+sqrt(10))^n + (3-sqrt(10))^n)/2.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k) * 10^k * 3^(n-2*k). (End)
a(n) = (-1)^n * a(-n) for all n in Z. - Michael Somos, Jul 14 2018 [This refers to the sequence extended to negative indices according to the recurrence relation, but not to the sequence as it is currently defined. - M. F. Hasler, Nov 02 2019]
a(n) = Lucas(n,6)/2, Lucas polynomial, L(n,x), evaluated at x = 6. - G. C. Greubel, Jun 06 2019

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

A040006 Continued fraction for sqrt(10).

Original entry on oeis.org

3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

N. J. A. Sloane

Eventual period is (6). - Zak Seidov, Mar 05 2011
The convergents are given in A005667(n+1)/A005668(n+1), n >= 0. - Wolfdieter Lang, Nov 23 2017
Decimal expansion of 11/30. - Elmo R. Oliveira, Feb 16 2024

			3.162277660168379331998893544... = 3 + 1/(6 + 1/(6 + 1/(6 + 1/(6 + ...)))).

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

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 (1).

Cf. A010467 (decimal expansion), A005667(n+1)/A005668(n+1) (convergents), A248239 (Egyptian fraction).

Cf. A040000.

[6-3*(Binomial(2*n,n) mod 2): n in [0..100]]; // Vincenzo Librandi, Jan 03 2016

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

ContinuedFraction[Sqrt[10],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PadRight[{3},120,{6}] (* Harvey P. Dale, Aug 25 2024 *)

contfrac(sqrt(10)) \\ For illustration.

A040006(n)=if(n,6,3) \\ M. F. Hasler, Nov 02 2019

a(n) = 3 + 3*sign(n). a(n) = 6, n > 0. - Wesley Ivan Hurt, Nov 01 2013
From Elmo R. Oliveira, Feb 16 2024: (Start)
G.f.: 3*(1+x)/(1-x).
E.g.f.: 6*exp(x) - 3.
a(n) = 3*A040000(n). (End)

A176398 Decimal expansion of 3+sqrt(10).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 16 2010

Continued fraction expansion of 3+sqrt(10) is A010722.
This is the shape of a 6-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011
c^n = c*A005668(n) + A005668(n-1). - Gary W. Adamson, Apr 04 2024

			6.16227766016837933199...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Wikipedia, Metallic mean.
Index entries for algebraic numbers, degree 2

Cf. A010467 (decimal expansion of sqrt(10)), A010722 (all 6's sequence).

Cf. A049310.

Digits:=100; evalf(3+sqrt(10)); # Wesley Ivan Hurt, Mar 07 2014

r=6; t=(r+(4+r^2)^(1/2))/2; RealDigits[N[t,130]][[1]] (* Clark Kimberling, Apr 09 2011 *)

3+sqrt(10) \\ Charles R Greathouse IV, Jul 24 2013

a(n) = A010467(n) for n >= 2.
Equals exp(arcsinh(3)), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals lim_{n->oo} S(n, 2*sqrt(10))/ S(n-1, 2*sqrt(10)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A017934 Powers of sqrt(10) rounded down.

Original entry on oeis.org

1, 3, 10, 31, 100, 316, 1000, 3162, 10000, 31622, 100000, 316227, 1000000, 3162277, 10000000, 31622776, 100000000, 316227766, 1000000000, 3162277660, 10000000000, 31622776601, 100000000000, 316227766016, 1000000000000, 3162277660168, 10000000000000

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A010467 Decimal expansion of sqrt(10), A131581, A136582, A175733, A175734.

A017934 := func< n | Isqrt(10^n) >; // Jason Kimberley, Aug 19 2011

A017934 := func< n | Floor(Sqrt(RealField(n)!10^n)) >; // Jason Kimberley, Aug 19 2011

Floor[Sqrt[10^Range[0,100]]] (* Zak Seidov *)

a(n)=sqrtint(10^n) \\ Charles R Greathouse IV, Nov 18 2011

a(n) = floor(sqrt(10^n)). - Zak Seidov

A010494 Decimal expansion of square root of 40.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			6.324555320336758663997787088865437067439110278650433653715009705585188....

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

Cf. A040033 (continued fraction). - Harry J. Smith, Jun 05 2009

Cf. A000045, A010467.

RealDigits[N[Sqrt[40],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

default(realprecision, 20080); x=sqrt(40); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010494.txt", n, " ", d));  \\ Harry J. Smith, Jun 05 2009

Equals 2*A010467. - R. J. Mathar, Jan 14 2021

Equals 10 * Sum_{k>=1} Fibonacci(k)*binomial(2*k,k)/8^k. - Amiram Eldar, Jan 17 2022

A020797 Decimal expansion of 1/sqrt(40).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

With offset 1, decimal expansion of sqrt(5/2). - Eric Desbiaux, May 01 2008
sqrt(5/2) appears as a coordinate in a degree-5 integration formula on 13 points in the unit sphere [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011
With offset 2, decimal expansion of sqrt(250). - Michel Marcus, Nov 04 2013
From Wolfdieter Lang, Nov 21 2017: (Start)
The regular continued fraction of 1/sqrt(40) = 1/(2*sqrt(10)) is [0; 6, 3, repeat(12, 3)], and the convergents are given by A(n-1)/B(n-1), n >= 0, with A(-1) = 0, A(n-1) = A041067(n) and B(-1) = 1, B(n-1) = A041066(n).
The regular continued fraction of sqrt(5/2) = sqrt(10)/2 is [1; repeat(1, 1, 2)], and the convergents are given in A295333/A295334.
sqrt(10)/2 is one of the catheti of the rectangular triangle with hypotenuse sqrt(13)/2 = A295330 and the other cathetus sqrt(3)/2 = A010527. This can be constructed from a regular hexagon inscribed in a circle with a radius of 1 unit. If the vertex V_0 has coordinates (x, y) = (1, 0) and the midpoint M_4 = (0, -sqrt(3)/2) then the point L = (sqrt(10)/2, 0) is obtained as intersection of the x-axis and a circle around M_4 with radius taken from the distance between M_4 and V_1 = (1/2, sqrt(3)/2) which is sqrt(13)/2. (End)

			1/sqrt(40) = 0.15811388300841896659994467722163592668597775696626084134287...
sqrt(5/2) = 1.5811388300841896659994467722163592668597775696626084134287...
sqrt(250) = 15.811388300841896659994467722163592668597775696626084134287...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
A. H. Stroud, D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Index entries for algebraic numbers, degree 2.

Cf. A010467 (sqrt(10)), A010527, A010494 (sqrt(40)), A041067/A041066, A295330, A295333/A295334.

RealDigits[N[1/Sqrt[40],200]] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)

1/sqrt(40) \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(40*x^2-1)[2] \\ Charles R Greathouse IV, Feb 04 2025

Equals Re(sqrt(5*i)/10) = Im(sqrt(5*i)/10). - Karl V. Keller, Jr., Sep 01 2020

Equals A010467/20. - R. J. Mathar, Feb 23 2021

A177102 Beatty sequence for sqrt(10).

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 22, 25, 28, 31, 34, 37, 41, 44, 47, 50, 53, 56, 60, 63, 66, 69, 72, 75, 79, 82, 85, 88, 91, 94, 98, 101, 104, 107, 110, 113, 117, 120, 123, 126, 129, 132, 135, 139, 142, 145, 148, 151, 154, 158, 161, 164, 167, 170, 173, 177, 180, 183

Offset: 1

Clark Kimberling, Aug 16 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A010467, A194113.

Partial sums of A081168.

[Floor(n*Sqrt(10)): n in [1..60]]; // Vincenzo Librandi, Oct 24 2011

Mathematica
```
Table[Floor[n*Sqrt[10]],{n,1,100}]
```

for(n=1,50, print1(floor(n*sqrt(10)), ", ")) \\ G. C. Greubel, Sep 24 2017

a(n) = floor(n*sqrt(10)).

A136582 Sqrt(10)-primes: primes obtained by concatenating the first digits in the decimal expansion of sqrt(10).

Original entry on oeis.org

3, 31, 3162277, 316227766016837933, 316227766016837933199889354443271

Offset: 1

Lekraj Beedassy, Jan 09 2008

n such that floor(sqrt(10^(2*n-1))) is prime (1, 2, 7, 18, 33,  ...) are given in A136583.
This sequence is the list of prime terms in A017934.
The next term has 206 digits. - Harvey P. Dale, Dec 06 2023

Cf. A010467, A017934, A017934, A131581, A132153, A136583, A175733, A175734.

// by Jason Kimberley, Aug 2011
for n in [1..499 by 2] do
  f := Isqrt(10^n);
  if IsPrime(f) then
    printf "%o,", f;
  end if;
end for;

Module[{nn=50,sq10},sq10=RealDigits[Sqrt[10],10,nn][[1]];Select[FromDigits/@Table[Take[sq10,n],{n,nn}],PrimeQ]] (* Harvey P. Dale, Dec 06 2023 *)

a(n) = A017934(2*A136583(n)-1).

A176221 Decimal expansion of sqrt(110).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 12 2010

Continued fraction expansion of sqrt(110) is A040099.

			sqrt(110) = 10.48808848170151546991...

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

Cf. A010467 (decimal expansion of sqrt(10)), A010468 (decimal expansion of sqrt(11)), A040099.

RealDigits[Sqrt[110], 10, 110][[1]] (* Alonso del Arte, Jan 06 2013 *)

PARI
```
sqrt(110) \\ Amiram Eldar, Aug 04 2022
```

Equals 10 * Sum_{k>=0} binomial(2*k,k)/44^k. - Amiram Eldar, Aug 04 2022

cp's OEIS Frontend

A248239 Egyptian fraction representation of sqrt(10) (A010467) using a greedy function.

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A005667 Numerators of continued fraction convergents to sqrt(10).

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A040006 Continued fraction for sqrt(10).

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A176398 Decimal expansion of 3+sqrt(10).

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A017934 Powers of sqrt(10) rounded down.

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A010494 Decimal expansion of square root of 40.

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A020797 Decimal expansion of 1/sqrt(40).

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