A040002 -id:A040002 - cp's OEIS frontend

A040380 Continued fraction for sqrt(401).

20, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40

Offset: 0

N. J. A. Sloane

			20 + 1/(40 + 1/(40 + 1/(40 + 1/(40 + ...)))) = sqrt(401).

Cf. A041760/A041761 (convergents).

Cf. A040000, A040002, A040012.

with(numtheory): Digits := 300: convert(evalf(sqrt(401)),confrac);

From Elmo R. Oliveira, Feb 15 2024: (Start)
a(n) = 40 for n >= 1.
G.f.: 20*(1+x)/(1-x).
E.g.f.: 40*exp(x) - 20.
a(n) = 20*A040000(n) = 10*A040002(n) = 5*A040012(n). (End)

A040462 Continued fraction for sqrt(485).

Original entry on oeis.org

22, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44

Offset: 0

N. J. A. Sloane

			22 + 1/(44 + 1/(44 + 1/(44 + 1/(44 + ...)))) = sqrt(485).

Cf. A041924/A041925 (convergents).

Cf. A040000, A040002, A040110.

with(numtheory): Digits := 300: convert(evalf(sqrt(485)),confrac);

From Elmo R. Oliveira, Feb 15 2024: (Start)
a(n) = 44 for n >= 1.
G.f.: 22*(1+x)/(1-x).
E.g.f.: 44*exp(x) - 22.
a(n) = 22*A040000(n) = 11*A040002(n) = 2*A040110(n). (End)

A040650 Continued fraction for sqrt(677).

Original entry on oeis.org

26, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52

Offset: 0

N. J. A. Sloane

			26 + 1/(52 + 1/(52 + 1/(52 + 1/(52 + ...)))) = sqrt(677).

Cf. A042300/A042301 (convergents).

Cf. A040000, A040002, A040156.

with(numtheory): Digits := 300: convert(evalf(sqrt(677)),confrac);

From Elmo R. Oliveira, Feb 15 2024: (Start)
a(n) = 52 for n >= 1.
G.f.: 26*(1+x)/(1-x).
E.g.f.: 52*exp(x) - 26.
a(n) = 26*A040000(n) = 13*A040002(n) = 2*A040156(n). (End)

A040756 Continued fraction for sqrt(785).

Original entry on oeis.org

28, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56

Offset: 0

N. J. A. Sloane

			28 + 1/(56 + 1/(56 + 1/(56 + 1/(56 + ...)))) = sqrt(785).

Cf. A042512/A042513 (convergents).

Cf. A040000, A040002, A040012.

with(numtheory): Digits := 300: convert(evalf(sqrt(785)),confrac);

Block[{$MaxExtraPrecision =1000},ContinuedFraction[Sqrt[785],70]] (* or *) PadRight[{28},70,56] (* Harvey P. Dale, May 09 2012 *)

From Elmo R. Oliveira, Feb 16 2024: (Start)
a(n) = 56 for n >= 1.
G.f.: 28*(1+x)/(1-x).
E.g.f.: 56*exp(x) - 28.
a(n) = 28*A040000(n) = 14*A040002(n) = 7*A040012(n). (End)

A255176 a(n) = H_n(2,2) where H_n is the n-th hyperoperator.

Original entry on oeis.org

3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Natan Arie Consigli, Feb 25 2015

See A054871 for definitions and key links.
Also, decimal expansion of 31/90. - Bruno Berselli, Mar 18 2015
Essentially the same as A010709, A040002, A113311, A123932, and A151798. - R. J. Mathar, Mar 20 2015
Remainder of the Euclidean division when 10^(10^n) is divided by 7 (proof by induction for n >= 1) [see reference Julien Freslon & Jérôme Poineau]; example: 10^(10^1) = 1428571428 * 7 + 4. - Bernard Schott, Aug 28 2020

			a(0) = H_0(2,2) = 2+1 = 3.
a(1) = H_1(2,2) = 2+2 = 4.
a(2) = H_2(2,2) = 2*2 = 4.
a(3) = H_3(2,2) = 2^2 = 4.
a(n) = H_n(2,2) = H_{n-1}(2,H_n(2,1)) = H_{n-1}(2,2) = 4, for n>1.

Julien Freslon & Jérôme Poineau, Les 100 exercices-types de mathématiques: MPSI/PCSI/PTSI, EdiScience, 2007, Exercice 11.2, page 242.

Wikipedia, Hyperoperation.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A054871.

Cf. A040000, A157532.

G.f.: (3 + x)/(1 - x). - Bruno Berselli, Mar 18 2015

a(n) = 10^(10^n) mod 7. - Bernard Schott, Aug 28 2020

Edited by Danny Rorabaugh, Oct 20 2015

A040090 Continued fraction for sqrt(101).

Original entry on oeis.org

10, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20

Offset: 0

N. J. A. Sloane

			10 + 1/(20 + 1/(20 + 1/(20 + 1/(20 + ...)))) = sqrt(101).

Cf. A248803 (decimal expansion), A041180/A041181 (convergents).

Cf. A040000, A040002, A040020.

Digits := 200: convert(evalf(sqrt(101)),confrac,150,'cvgts'):

ContinuedFraction[Sqrt[101],300] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2011*)
PadRight[{10},100,{20}] (* Harvey P. Dale, Apr 22 2022 *)

From Elmo R. Oliveira, Feb 11 2024: (Start)
a(n) = 20 = A010859(n) for n >= 1.
G.f.: 10*(1+x)/(1-x).
E.g.f.: 20*exp(x) - 10.
a(n) = 10*A040000(n) = 5*A040002(n) = 2*A040020(n). (End)

A040182 Continued fraction for sqrt(197).

Original entry on oeis.org

14, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28

Offset: 0

N. J. A. Sloane

			14 + 1/(28 + 1/(28 + 1/(28 + 1/(28 + ...)))) = sqrt(197).

Cf. A041364/A041365 (convergents).

Cf. A040000, A040002, A040042.

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

ContinuedFraction[Sqrt[197],300] (* Vladimir Joseph Stephan Orlovsky, Mar 28 2011 *)
PadRight[{14},120,{28}] (* Harvey P. Dale, May 02 2022 *)

From Elmo R. Oliveira, Feb 13 2024: (Start)
a(n) = 28 for n >= 1.
G.f.: 14*(1+x)/(1-x).
E.g.f.: 28*exp(x) - 14.
a(n) = 14*A040000(n) = 7*A040002(n) = 2*A040042(n). (End)

A040240 Continued fraction for sqrt(257).

Original entry on oeis.org

16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32

Offset: 0

N. J. A. Sloane

			16 + 1/(32 + 1/(32 + 1/(32 + 1/(32 + ...)))) = sqrt(257).

Cf. A041480/A041481 (convergents).

Cf. A040000, A040002, A040012, A040056.

with(numtheory): Digits := 300: convert(evalf(sqrt(257)),confrac);

Block[{$MaxExtraPrecision=1000}, ContinuedFraction[Sqrt[257],100]] (* or *) PadRight[{16},100,{32}] (* Harvey P. Dale, Aug 15 2021 *)

a(n)=if(n,32,16) \\ Charles R Greathouse IV, Apr 08 2012

From Elmo R. Oliveira, Feb 13 2024: (Start)
a(n) = 32 for n >= 1.
G.f.: 16*(1+x)/(1-x).
E.g.f.: 32*exp(x) - 16.
a(n) = 16*A040000(n) = 8*A040002(n) = 4*A040012(n) = 2*A040056(n). (End)

A040552 Continued fraction for sqrt(577).

Original entry on oeis.org

24, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48

Offset: 0

N. J. A. Sloane

			24 + 1/(48 + 1/(48 + 1/(48 + 1/(48 + ...)))) = sqrt(577).

Cf. A042104/A042105 (convergents).

Cf. A040000, A040002, A040006.

with(numtheory): Digits := 300: convert(evalf(sqrt(577)),confrac);

From Elmo R. Oliveira, Feb 15 2024: (Start)
a(n) = 48 for n >= 1.
G.f.: 24*(1+x)/(1-x).
E.g.f.: 48*exp(x) - 24.
a(n) = 24*A040000(n) = 12*A040002(n) = 8*A040006(n). (End)

A040930 Continued fraction for sqrt(962).

Original entry on oeis.org

31, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62

Offset: 0

N. J. A. Sloane

			31 + 1/(62 + 1/(62 + 1/(62 + 1/(62 + ...)))) = sqrt(962).

Cf. A042860/A042861 (convergents).

Continued fraction for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040000 (contfrac(sqrt(2)) = (1,2,2,...)), A040002, A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870 (contfrac(sqrt(901)) = (30,60,60,...)).

with(numtheory): Digits := 300: convert(evalf(sqrt(962)),confrac);

PadRight[{31},100,62] (* Harvey P. Dale, Sep 18 2012 *)

G.f.: 31*(1+x)/(1-x). - Colin Barker, Aug 11 2012
From Elmo R. Oliveira, Feb 16 2024: (Start)
a(n) = 62 for n >= 1.
E.g.f.: 62*exp(x) - 31.
a(n) = 31*A040000(n). (End)

cp's OEIS Frontend

A040380 Continued fraction for sqrt(401).

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A040462 Continued fraction for sqrt(485).

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A040650 Continued fraction for sqrt(677).

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A040756 Continued fraction for sqrt(785).

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A255176 a(n) = H_n(2,2) where H_n is the n-th hyperoperator.

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A040090 Continued fraction for sqrt(101).

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A040182 Continued fraction for sqrt(197).

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A040240 Continued fraction for sqrt(257).

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