A007400 -id:A007400 - cp's OEIS frontend

A010148 Continued fraction for sqrt(69).

8, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1

Offset: 0

N. J. A. Sloane

			8.306623862918074852584262744... = 8 + 1/(3 + 1/(3 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 08 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A010521 Decimal expansion. - Harry J. Smith, Jun 08 2009

ContinuedFraction[Sqrt[69],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16,3,3,1,4,1,3,3}] (* Harvey P. Dale, Jan 25 2024 *)

{ allocatemem(932245000); default(realprecision, 22000); x=contfrac(sqrt(69)); for (n=0, 20000, write("b010148.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 08 2009

A076157 Continued fraction expansion for c=sum_{k>=0} 1/2^(k!).

Original entry on oeis.org

1, 3, 1, 3, 4, 4095, 1, 3, 3, 1, 3, 4722366482869645213695, 1, 2, 1, 3, 3, 1, 4095, 4, 3, 1, 3, 3121748550315992231381597229793166305748598142664971150859156959625371738819765620120306103063491971159826931121406622895447975679288285306290175

Offset: 1

Benoit Cloitre, Nov 02 2002

Observation: if b(k) denotes the sequence of all elements of the continued fraction for c, b(k) = 4095 if k==6 or 19 (mod 24); b(k) = 4722366482869645213695 if k==12 or 37 (mod 48); .... If b(k) is not congruent to 5 (mod 10), it seems that b(k) = 1,2,3 or 4 only.
Conjecture: a(3*2^n) = -1 + 2^[(n+1)((n+2)!) ]. - Ralf Stephan, May 17 2005
The conjecture follows from the theorem in Shallit's paper. The continued fraction has a "folded" overall structure. - Georg Fischer, Aug 29 2022

Rick L. Shepherd, Table of n, a(n) for n = 1..47
Jeffrey O. Shallit, Simple Continued Fractions for Some Irrational Numbers II, J. Number Theory 14 (1982), 228-231.
Rick L. Shepherd, Table of n, a(n) for n = 1..383 (has larger terms than b-files support)

Cf. A076152, A076154, A076187.

Cf. A007400, A004200, A006466.

{allocatemem(220000000);
default(realprecision, 1000000);
contfrac(suminf(k=0, 1/(2^(k!))))}

c=1.2656250596046447753906250000000000007... = A076187.

More terms from Ralf Stephan, May 17 2005

b-file, a-file, PARI program, and corrected conjecture by Rick L. Shepherd, Jun 07 2013

A010136 Continued fraction for sqrt(46).

Original entry on oeis.org

6, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 1, 1, 2

Offset: 0

N. J. A. Sloane

			6.782329983125268139064556326... = 6 + 1/(1 + 1/(3 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 06 2009

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
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A010500 (decimal expansion), A041078/A041079 (convergents), A248272 (Egyptian fractions).

Cf. A190567.

ContinuedFraction[Sqrt[46],300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(46)); for (n=0, 20000, write("b010136.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 06 2009

A041078 Numerators of continued fraction convergents to sqrt(46).

Original entry on oeis.org

6, 7, 27, 34, 61, 156, 997, 2150, 3147, 5297, 19038, 24335, 311058, 335393, 1317237, 1652630, 2969867, 7592364, 48524051, 104640466, 153164517, 257804983, 926579466, 1184384449, 15139192854, 16323577303

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,48670,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A010500, A041079.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[46],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011 *)
Numerator[Convergents[Sqrt[46], 30]] (* Vincenzo Librandi, Oct 25 2013 *)

a(n) = 48670*a(n-12)-a(n-24). G.f.: -(x^23 -6*x^22 +7*x^21 -27*x^20 +34*x^19 -61*x^18 +156*x^17 -997*x^16 +2150*x^15 -3147*x^14 +5297*x^13 -19038*x^12 -24335*x^11 -19038*x^10 -5297*x^9 -3147*x^8 -2150*x^7 -997*x^6 -156*x^5 -61*x^4 -34*x^3 -27*x^2 -7*x-6) / (x^24-48670*x^12+1). - Colin Barker, Jul 19 2012

Formula corrected by Colin Barker, Jul 24 2012

A041079 Denominators of continued fraction convergents to sqrt(46).

Original entry on oeis.org

1, 1, 4, 5, 9, 23, 147, 317, 464, 781, 2807, 3588, 45863, 49451, 194216, 243667, 437883, 1119433, 7154481, 15428395, 22582876, 38011271, 136616689, 174627960, 2232152209, 2406780169, 9452492716, 11859272885

Offset: 0

N. J. A. Sloane

46 is the smallest value of n for which the period of the continued fraction convergents to sqrt(n) is 12. [Colin Barker, Jul 19 2012]

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,48670,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A010500, A041078.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[46],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011*)
Denominator[Convergents[Sqrt[46], 30]] (* Vincenzo Librandi, Oct 24 2013 *)

a(n) = 48670*a(n-12)-a(n-24). G.f.: -(x^22 -x^21 +4*x^20 -5*x^19 +9*x^18 -23*x^17 +147*x^16 -317*x^15 +464*x^14 -781*x^13 +2807*x^12 -3588*x^11 -2807*x^10 -781*x^9 -464*x^8 -317*x^7 -147*x^6 -23*x^5 -9*x^4 -5*x^3 -4*x^2 -x -1) / (x^24-48670*x^12+1). [Colin Barker, Jul 19 2012]

Formula corrected by Colin Barker, Jul 24 2012

A073414 Numerator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

Original entry on oeis.org

0, 1, 4, 9, 40, 169, 1054, 4385, 9824, 43681, 271910, 587501, 2621914, 16318985, 67897854, 287910401, 643718656, 2862785025, 17820428806, 38503642637, 171834999354, 725843640053, 4526896839672, 18833430998741, 42193758837154

Offset: 1

Benoit Cloitre, Aug 23 2002

Robert Israel, Table of n, a(n) for n = 1..1650

Cf. A007400, A007404, A073415.

a007400:= proc(n) option remember; local n8, n16;
    n8:= n mod 8;
    if n8 = 0 or n8 = 3 then return 2
    elif n8 = 4 or n8 = 7 then return 4
    elif n8 = 1 then return procname((n+1)/2)
    elif n8 = 2 then return procname((n+2)/2)
    fi;
    n16:= n mod 16;
    if n16 = 5 or n16 = 14 then return 4
    elif n16 = 6 or n16 = 13 then return 6
    fi
end proc:
a007400(0):= 0: a007400(1):= 1: a007400(2):= 4:
A[1]:= 0: A[2]:= 1:
for n from 3 to 100 do
  A[n]:= A[n-1]*a007400(n-1)+A[n-2];
od:
seq(A[n],n=1..100); # Robert Israel, Jun 14 2016

(* b is a007400 *)
b[n_] := b[n] = Module[{n8, n16}, n8 = Mod[n, 8]; Which[n8 == 0 || n8 == 3, Return[2], n8 == 4 || n8 == 7, Return[4], n8 == 1, Return[b[(n+1)/2]], n8 == 2, Return[b[(n+2)/2]]]; n16 = Mod[n, 16]; Which[n16 == 5 || n16 == 14, Return[4], n16 == 6 || n16 == 13, Return[6]]];
b[0] = 0; b[1] = 1; b[2] = 4;
a[1] = 0; a[2] = 1;
a[n_] := a[n] = a[n-1] b[n-1] + a[n-2];
Array[a, 100] (* Jean-François Alcover, Jun 10 2020, after Robert Israel *)

a(n)=component(component(contfracpnqn(contfrac(sum(k=0,20,1/2^(2^k)),n)),1),1)

a(n) = a(n-1) A007400(n-1) + a(n-2). - Robert Israel, Jun 14 2016

A073415 Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

Original entry on oeis.org

1, 1, 5, 11, 49, 207, 1291, 5371, 12033, 53503, 333051, 719605, 3211471, 19988431, 83165195, 352649211, 788463617, 3506503679, 21827485691, 47161475061, 210473385935, 889055018801, 5544803498741, 23068269013765, 51681341526271

Offset: 1

Benoit Cloitre, Aug 23 2002

Robert Israel, Table of n, a(n) for n = 1..1650

Cf. A007400, A007404, A073414.

a007400:= proc(n) option remember; local n8, n16;
    n8:= n mod 8;
    if n8 = 0 or n8 = 3 then return 2
    elif n8 = 4 or n8 = 7 then return 4
    elif n8 = 1 then return procname((n+1)/2)
    elif n8 = 2 then return procname((n+2)/2)
    fi;
    n16:= n mod 16;
    if n16 = 5 or n16 = 14 then return 4
    elif n16 = 6 or n16 = 13 then return 6
    fi
end proc:
a007400(0):= 0: a007400(1):= 1: a007400(2):= 4:
A[1]:= 1: A[2]:= 1:
for n from 3 to 100 do
  A[n]:= A[n-1]*a007400(n-1)+A[n-2];
od:
seq(A[n], n=1..100); # Robert Israel, Jun 14 2016

(* b is a007400 *)
b[n_] := b[n] = Module[{n8, n16}, n8 = Mod[n, 8]; Which[n8 == 0 || n8 == 3, Return[2], n8 == 4 || n8 == 7, Return[4], n8 == 1, Return[b[(n+1)/2]], n8 == 2, Return[b[(n+2)/2]]]; n16 = Mod[n, 16]; Which[n16 == 5 || n16 == 14, Return[4], n16 == 6 || n16 == 13, Return[6]]];
b[0] = 0; b[1] = 1; b[2] = 4;
a[1] = a[2] = 1;
a[n_] := a[n] = a[n-1] b[n-1] + a[n-2];
Array[a, 100] (* Jean-François Alcover, Jun 10 2020, after Robert Israel *)

a(n)=component(component(contfracpnqn(contfrac(sum(k=0,20,1/2^(2^k)),n)),1),2)

a(n) = a(n-1) A007400(n) + a(n-2). - Robert Israel, Jun 14 2016

A092910 a(n) is the (3n+2)-th component of the continued fraction for sum(k>=0,2^(-k!)).

Original entry on oeis.org

3, 4, 3, 3, 2, 3, 4, 3, 2, 4, 3, 2, 3, 3, 4, 3, 2, 4, 3, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 3, 4, 3, 2, 4, 3, 3, 2, 3, 4, 3, 2, 4, 3, 2, 3, 3, 4, 2, 3, 4, 3, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 3, 4, 3, 2, 4, 3, 3, 2, 3, 4, 3, 2, 4, 3, 2, 3, 3, 4, 3, 2, 4, 3, 3, 2, 3, 4, 2, 3, 4, 3, 2, 3, 3, 4, 2, 3, 4, 3, 3, 2, 3, 4, 3, 2

Offset: 0

Benoit Cloitre, Apr 16 2004

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A007400, A076157, A088431.

a(n)=5-component(contfrac(sum(i=0,10,1/2^(2^i))),n+3)/2

(define (A092910 n) (- 5 (* 1/2 (A007400 (+ 2 n))))) ;;  Code for A007400 given under that entry. - Antti Karttunen, Aug 12 2017

a(n) = 5 - (A007400(n+2)/2).

A081846 Maximal element in the continued fraction for 1/n*sum(k>=0,1/2^(2^k)).

Original entry on oeis.org

6, 12, 19, 25, 33, 39, 46, 52, 60, 66, 72, 79, 85, 93, 99, 106, 112, 120, 126, 132, 139, 145, 153, 159, 166, 172, 180, 186, 192, 199, 205, 206, 219, 226, 232, 240, 246, 252, 259, 265, 273, 279, 286, 292, 300, 306, 313, 319, 326, 333, 339, 346, 352, 360, 366, 373, 379, 386, 393, 399, 406, 412, 420, 413, 433, 439, 446, 453, 459, 466, 472, 480, 486, 493, 499, 506, 513, 519, 526, 532, 540, 546, 553, 559, 567, 573, 579, 586, 593, 600, 606, 613, 619, 627, 633, 619, 646, 653, 660, 666

Offset: 1

Benoit Cloitre, Apr 10 2003

nonn

It seems that a(n)=20n/3 for infinitely many values of n.

Cf. A007400, A007404, A078816 (erroneous version).

Cf. A384939.

s = N[Sum[1/2^(2^k), {k, 0, Infinity}], 1000000]; Table[Max[ContinuedFraction[s/n]], {n, 1, 100}] (* Vaclav Kotesovec, Jul 22 2025 *)

Corrected and extended by Vaclav Kotesovec, Jul 22 2025

A089267 Continued fraction expansion with iterated 3-fold symmetry.

Original entry on oeis.org

0, 1, 1, 23, 1, 2, 1, 18815, 3, 1, 23, 3, 1, 23, 1, 2, 1, 106597754640383, 3, 1, 23, 1, 3, 23, 1, 3, 18815, 1, 2, 1, 23, 3, 1, 23, 1, 2, 1, 18815, 3, 1, 23, 3, 1, 23, 1, 2, 1, 1715738475058821295603924428015888899408203312889855, 3, 1

Offset: 1

Ralf Stephan, Oct 30 2003

H. Cohn, Symmetry and specializability in continued fractions

Cf. A007400.

nmax = 50; f[m_] := ContinuedFraction[ Sum[ 1/ChebyshevT[4^k, 2], {k, 0, m}]]; A089267 = Catch[ For[m = 1, True, m++, If[ Length[fm = f[m]] > nmax, Throw[ fm[[1 ;; nmax]] ]]]] (* Jean-François Alcover, Sep 19 2012 *)

contfrac(suminf(k=0,1/subst(poltchebi(4^k),x,2)))

contfrac(suminf(k=0,1/polchebyshev(4^k,1,2))) \\ Charles R Greathouse IV, May 28 2015

Sum_{k=0..infinity} 1/chebyshev(4^k, 2) = 0.51030927976262776140...

cp's OEIS Frontend

A010148 Continued fraction for sqrt(69).

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A076157 Continued fraction expansion for c=sum_{k>=0} 1/2^(k!).

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A010136 Continued fraction for sqrt(46).

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A041078 Numerators of continued fraction convergents to sqrt(46).

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A041079 Denominators of continued fraction convergents to sqrt(46).

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A073414 Numerator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

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Maple

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A073415 Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

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Maple

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A092910 a(n) is the (3n+2)-th component of the continued fraction for sum(k>=0,2^(-k!)).

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