A010468 -id:A010468 - cp's OEIS frontend

A194389 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(11) and < > denotes fractional part.

1, 17, 19, 20, 21, 23, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 55, 57, 58, 59, 61, 77, 79, 80, 81, 83, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 115, 117, 118, 119, 121, 137, 139, 140, 141, 143, 153, 155, 156, 157, 158, 159, 160, 161, 162, 163

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010468, A194368, A194387, A194388.

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 *)

A040007 Continued fraction for sqrt(11).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Eventual period is (3,6). - Zak Seidov, Mar 05 2011

Decimal expansion of 37/110. - R. J. Mathar, Aug 22 2025

			3.316624790355399849114932736... = 3 + 1/(3 + 1/(6 + 1/(3 + 1/(6 + ...)))). - _Harry J. Smith_, Jun 02 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, 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 (0,1).

Cf. A010468, A010704.

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

ContinuedFraction[Sqrt[11],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{3},120,{6,3}] (* Harvey P. Dale, Jan 18 2025 *)

{ allocatemem(932245000); default(realprecision, 27000); x=contfrac(sqrt(11)); for (n=0, 20000, write("b040007.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

From Stefano Spezia, Jan 18 2025: (Start)
G.f.: 3*(1 + x + x^2)/(1 - x^2).
E.g.f.: 3*(2*cosh(x) + sinh(x) - 1). (End)

A176395 Decimal expansion of 3+sqrt(11).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 16 2010

Continued fraction expansion of 3+sqrt(11) is A010704 preceded by 6.

a(n) = A010468(n) for n > 1.

			3+sqrt(11) = 6.31662479035539984911...

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

Cf. A010468 (decimal expansion of sqrt(11)), A010704 (repeat 3, 6).

RealDigits[3+Sqrt[11],10,120][[1]] (* Harvey P. Dale, Oct 28 2011 *)

A083334 a(n) = 12a(n-2) - 25a(n-4) with initial terms 1,6,17,47.

Original entry on oeis.org

1, 6, 17, 47, 179, 414, 1723, 3793, 16201, 35166, 151337, 327167, 1411019, 3046854, 13148803, 28383073, 122510161, 264425526, 1141401857, 2463529487, 10634068259, 22951715694, 99073772683, 213832351153, 923033565721

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 26 2003

a(n)/A083335(n) converges to sqrt(11).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-25).

Cf. A010468 (sqrt(11)), A083335.

CoefficientList[Series[(1+6x+5x^2-25x^3)/(1-12x^2+25x^4), {x, 0, 10}], x]
LinearRecurrence[{0,12,0,-25},{1,6,17,47},30] (* Harvey P. Dale, Oct 15 2012 *)

G.f.: (1 + 6*x + 5*x^2 - 25*x^3) / (1 - 12*x^2 + 25*x^4).

a(n) = (x^(n+1) + (-1)^(n+1)*(x-2)^(n+1))/2^ceiling(n/2 + 1) where x = 1 + sqrt(11). - Ben Paul Thurston, Aug 30 2006 [edited by Jon E. Schoenfield, Jun 25 2019]

A147313 Decimal expansion of sqrt(11)/2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 22 2009

			1.6583123951776999245574663683353433419635442727946767985293...

D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 308.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A010468 (sqrt(11)).

SetDefaultRealField(RealField(120)); Sqrt(11)/2; // G. C. Greubel, Jan 09 2020

evalf( sqrt(11)/2, 120); # G. C. Greubel, Jan 09 2020

RealDigits[Sqrt[11]/2,10,120][[1]] (* Harvey P. Dale, Mar 06 2012 *)

default(realprecision, 120); sqrt(11)/2 \\ G. C. Greubel, Jan 09 2020

numerical_approx(sqrt(11)/2, digits=120) # G. C. Greubel, Jan 09 2020

A176056 Decimal expansion of (3+sqrt(11))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (3+sqrt(11))/3 is A010699.

			(3+sqrt(11))/3 = 2.10554159678513328303...

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

Cf. A010468 (decimal expansion of sqrt(11)), A010699 (repeat 2, 9).

RealDigits[(3+Sqrt[11])/3,10,120][[1]] (* Harvey P. Dale, May 22 2015 *)

A176105 Decimal expansion of (3+sqrt(11))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 10 2010

Continued fraction expansion of (3+sqrt(11))/2 is A010704.

			(3+sqrt(11))/2 = 3.15831239517769992455...

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

Cf. A010468 (decimal expansion of sqrt(11)), A010704 (repeat 3, 6).

RealDigits[(3+Sqrt[11])/2,10,120][[1]] (* Harvey P. Dale, Apr 16 2013 *)

Equals 1.5 + A147313. [R. J. Mathar, Apr 12 2010]

A177839 Decimal expansion of sqrt(2442).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, May 14 2010

Continued fraction expansion of sqrt(2442) is 49 followed by (repeat 2, 2, 2, 98).

sqrt(2442) = sqrt(2)*sqrt(3)*sqrt(11)*sqrt(37).

			sqrt(2442) = 49.41659640242334617762...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000

Cf. A002193 (decimal expansion of sqrt(2)), A002194 (decimal expansion of sqrt(3)), A010468 (decimal expansion of sqrt(11)), A010491 (decimal expansion of sqrt(37)), A177838 (decimal expansion of (44+sqrt(2442))/88).

RealDigits[Sqrt[2442],10,120][[1]] (* Harvey P. Dale, Apr 01 2012 *)

A274526 a(n) = ((1 + sqrt(11))^n - (1 - sqrt(11))^n)/sqrt(11).

Original entry on oeis.org

0, 2, 4, 28, 96, 472, 1904, 8528, 36096, 157472, 675904, 2926528, 12612096, 54489472, 235099904, 1015094528, 4381188096, 18913321472, 81638523904, 352410262528, 1521205764096, 6566514153472, 28345085947904, 122355313430528, 528161486340096

Offset: 0

Ilya Gutkovskiy, Jun 27 2016

Number of zeros in substitution system {0 -> 11111, 1 -> 1001} at step n from initial string "1" (see example).

			Evolution from initial string "1": 1 -> 1001 -> 100111111111111001 -> 1001111111111110011001100110011001100110011001100110011001100111111111111001 -> ...
Therefore, number of zeros at step n:
a(0) = 0;
a(1) = 2;
a(2) = 4;
a(3) = 28, etc.

Ilya Gutkovskiy, Illustration (substitution system {0 -> 11111, 1 -> 1001})
Eric Weisstein's World of Mathematics, Substitution System
Index entries for linear recurrences with constant coefficients, signature (2,10)

Cf. A010468, A083102.

Mathematica
```
LinearRecurrence[{2, 10}, {0, 2}, 25]
```

concat(0, Vec(2*x/(1-2*x-10*x^2) + O(x^99))) \\ Altug Alkan, Jun 27 2016

O.g.f.: 2*x/(1 - 2*x - 10*x^2).
E.g.f.: 2*exp(x)*sinh(sqrt(11)*x)/sqrt(11).
Dirichlet g.f.: (PolyLog(s,1+sqrt(11)) - PolyLog(s,1-sqrt(11)))/sqrt(11), where PolyLog(s,x) is the polylogarithm function.
a(n) = 2*a(n-1) + 10*a(n-2).
a(n) = 2*A083102(n-1), n>0.
Lim_{n->infinity} a(n+1)/a(n) = 1 + sqrt(11) = 1 + A010468.

A017937 Powers of sqrt(11) rounded down.

Original entry on oeis.org

1, 3, 11, 36, 121, 401, 1331, 4414, 14641, 48558, 161051, 534145, 1771561, 5875603, 19487171, 64631634, 214358881, 710947978, 2357947691, 7820427766, 25937424601, 86024705429, 285311670611

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A010468 (sqrt(11)).

[Floor(Sqrt(11^n)): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Floor[(Sqrt[11])^Range[0,30]] (* Harvey P. Dale, Feb 01 2014 *)

a(n)=sqrtint(11^n) \\ Charles R Greathouse IV, Nov 18 2011

a(n) = floor(sqrt(11^n)). - Vincenzo Librandi, Jun 24 2011

cp's OEIS Frontend

A194389 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(11) and < > denotes fractional part.

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A040007 Continued fraction for sqrt(11).

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A176395 Decimal expansion of 3+sqrt(11).

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A083334 a(n) = 12*a(n-2) - 25*a(n-4) with initial terms 1,6,17,47.

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A147313 Decimal expansion of sqrt(11)/2.

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A176056 Decimal expansion of (3+sqrt(11))/3.

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A176105 Decimal expansion of (3+sqrt(11))/2.

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A177839 Decimal expansion of sqrt(2442).

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A083334 a(n) = 12a(n-2) - 25a(n-4) with initial terms 1,6,17,47.