A010468 -id:A010468 - cp's OEIS frontend

A248240 Egyptian fraction representation of sqrt(11) (A010468) using a greedy function.

3, 4, 16, 243, 104559, 25176928409, 26586186736052347315834, 1862816215759124563815793524962166009780011752, 5214712907768239185916350444296489272388117885310572145230445264540008760076034857528421553

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[ 11]]

A041014 Numerators of continued fraction convergents to sqrt(11).

Original entry on oeis.org

3, 10, 63, 199, 1257, 3970, 25077, 79201, 500283, 1580050, 9980583, 31521799, 199111377, 628855930, 3972246957, 12545596801, 79245827763, 250283080090, 1580944308303, 4993116004999, 31539640338297

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 (0,20,0,-1).

Cf. A010468, A041015 (denominators).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041016 (m=12), ..., A042936 (m=1000).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[11],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
Numerator[Convergents[Sqrt[11], 30]] (* Vincenzo Librandi, Oct 28 2013 *)

A041014=contfracpnqn(c=contfrac(sqrt(11)), #c)[1,][^-1] \\ Discard last element which may be incorrect. Use e.g. \p999 to get more terms, or extend as follows:
{A041014_upto(N,A=Vec(A041014,N))=for(n=#A041014+1,N, A[n]=20*A[n-2]-A[n-4]); A041014=A} \\ M. F. Hasler, Nov 01 2019

G.f.: (3 + 10*x + 3*x^2 - x^3)/(1 - 20*x^2 + x^4).

A041015 Denominators of continued fraction convergents to sqrt(11).

Original entry on oeis.org

1, 3, 19, 60, 379, 1197, 7561, 23880, 150841, 476403, 3009259, 9504180, 60034339, 189607197, 1197677521, 3782639760, 23893516081, 75463188003, 476672644099, 1505481120300, 9509559365899, 30034159217997

Offset: 0

N. J. A. Sloane

Sqrt(11) = 3 + continued fraction [3, 6, 3, 6, 3, 6, ...] = 6/2 + 6/19 + 6/(19*379) + 6/(379*7561) + ... - Gary W. Adamson, Dec 21 2007

Let X = the 2 X 2 matrix [1, 6; 3, 19], then X^n * [1, 0] = [a(n+1), a(n+2)]; e.g., X^3 * [1, 0] = [379, 1197] = [a(4), a(5)]. - Gary W. Adamson, Dec 21 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

Cf. A010468, A041014.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[11],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
a0[n_] := (11+3*Sqrt[11]+(11-3*Sqrt[11])*(10+3*Sqrt[11])^(2*n))/(22*(10+3*Sqrt[11])^n) // Simplify
a1[n_] := 3*Sum[a0[i], {i, 1, n}]
Flatten[MapIndexed[{a0[#], a1[#]}&,Range[11]]] (* Gerry Martens, Jul 10 2015 *)

G.f.: (1+3*x-x^2)/(1-20*x^2+x^4). - Colin Barker, Dec 31 2011
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)]:
a0(n) = ((11+3*sqrt(11))/(10+3*sqrt(11))^n + (11-3*sqrt(11))*(10+3*sqrt(11))^n)/22.
a1(n) = 3*Sum_{i=1..n} a0(i). (End)

A378204 Decimal expansion of the surface area of a triakis tetrahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 20 2024

The triakis tetrahedron is the dual polyhedron of the truncated tetrahedron.

			5.5277079839256664151915545611178111398784809093...

Eric Weisstein's World of Mathematics, Triakis Tetrahedron.
Wikipedia, Triakis tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A378205 (volume), A378206 (inradius), A378207 (midradius), A378208 (dihedral angle).
Cf. A377274 (surface area of a truncated tetrahedron with unit edge).
Cf. A010468.

First[RealDigits[5*Sqrt[11]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisTetrahedron", "SurfaceArea"], 10, 100]]

Equals (5/3)*sqrt(11) = (5/3)*A010468.

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

A010498 Decimal expansion of square root of 44.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 6 followed by {1, 1, 1, 2, 1, 1, 1, 12} repeated. [Harry J. Smith, Jun 05 2009]

			6.633249580710799698229865473341373367854177091178707194117364292232969...

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

Cf. A040037 (continued fraction).

RealDigits[N[Sqrt[44],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

{ default(realprecision, 20080); x=sqrt(44); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010498.txt", n, " ", d)); } \\ Harry J. Smith, Jun 05 2009

Equals 2*A010468. - R. J. Mathar, Jan 14 2021

A123482 Coefficients of the series giving the best rational approximations to sqrt(11).

Original entry on oeis.org

60, 23940, 9528120, 3792167880, 1509273288180, 600686976527820, 239071907384784240, 95150018452167599760, 37869468272055319920300, 15071953222259565160679700, 5998599512991034878630600360, 2387427534217209622129818263640, 950190160018936438572789038328420

Offset: 1

Gene Ward Smith, Oct 02 2006

The partial sums of the series 10/3 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(11), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [3;3,6,3], [3;3,6,3,6,3], [3;3,6,3,6,3,6,3] and so forth.

G. C. Greubel, Table of n, a(n) for n = 1..380
Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

Cf. A010468, A041014, A041015.

Cf. A123478, A123479, A123480, A029549.

CoefficientList[Series[-60*x/((x - 1)*(x^2 - 398*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 13 2017 *)

Vec(-60*x/((x-1)*(x^2-398*x+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

a(n+3) = 399*a(n+2) - 399*a(n+1) + a(n).
a(n) = -5/33 + (5/66 + 1/44*11^(1/2))*(199 + 60*11^(1/2))^n + (5/66 - 1/44*11^(1/2))*(199 - 60*11^(1/2))^n.
G.f.: -60*x / ((x-1)*(x^2-398*x+1)). - Colin Barker, Jun 23 2014

More terms from Colin Barker, Jun 23 2014

A177934 Decimal expansion of sqrt(71216963807).

Original entry on oeis.org

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

Offset: 6

Klaus Brockhaus, May 15 2010

Continued fraction expansion of sqrt(71216963807) is 266865 followed by (repeat 15, 533730).

sqrt(71216963807) = sqrt(11)*sqrt(19)*sqrt(107)*sqrt(179)*sqrt(17791).

			sqrt(71216963807) = 266865.06666665833952872396...

Cf. A010468 (decimal expansion of sqrt(11)), A010475 (decimal expansion of sqrt(19)), A177935 (decimal expansion of sqrt(107)), A177936 (decimal expansion of sqrt(179)), A177937 (decimal expansion of sqrt(17791)), A177933 (decimal expansion of (232405+sqrt(71216963807))/348378).

RealDigits[Sqrt[71216963807],10,120][[1]] (* Harvey P. Dale, Jul 31 2021 *)

A194387 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(11) and < > denotes fractional part.

Original entry on oeis.org

3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 25, 27, 28, 29, 31, 47, 49, 50, 51, 53, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 85, 87, 88, 89, 91, 107, 109, 110, 111, 113, 123, 125, 126, 127, 128, 129, 130, 131, 132, 133, 135, 145, 147, 148, 149, 151, 167, 169, 170

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010468, A194368, A194388, A194389.

r = Sqrt[11]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t1, 1]]     (* A194387 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t2, 1]]     (* A194388 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t3, 1]]     (* A194389 *)

A194388 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(11) and < > denotes fractional part.

Original entry on oeis.org

2, 4, 14, 16, 18, 22, 24, 26, 30, 32, 34, 44, 46, 48, 52, 54, 56, 60, 62, 64, 74, 76, 78, 82, 84, 86, 90, 92, 94, 104, 106, 108, 112, 114, 116, 120, 122, 124, 134, 136, 138, 142, 144, 146, 150, 152, 154, 164, 166, 168, 172, 174, 176, 180, 182, 184, 194, 196

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

Every term is even; see A194368.

Cf. A010468, A194368, A194387, A194389.

r = Sqrt[11]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t1, 1]]     (* A194387 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t2, 1]]     (* A194388 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t3, 1]]     (* A194389 *)

cp's OEIS Frontend

A248240 Egyptian fraction representation of sqrt(11) (A010468) using a greedy function.

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A041014 Numerators of continued fraction convergents to sqrt(11).

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A041015 Denominators of continued fraction convergents to sqrt(11).

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A378204 Decimal expansion of the surface area of a triakis tetrahedron with unit shorter edge length.

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A176221 Decimal expansion of sqrt(110).

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A010498 Decimal expansion of square root of 44.

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A123482 Coefficients of the series giving the best rational approximations to sqrt(11).

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A177934 Decimal expansion of sqrt(71216963807).

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