A040005 -id:A040005 - cp's OEIS frontend

A010466 Decimal expansion of square root of 8.

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

Offset: 1

N. J. A. Sloane

Sqrt(8) = 2*sqrt(2) is the length of the longest (rigid) ladder that can be carried horizontally around a right angled corner in a hallway of unit width. - Lekraj Beedassy, Apr 19 2006
Continued fraction expansion is 2 followed by {1, 4} repeated. - Harry J. Smith, Jun 05 2009
This is the second Lagrange number. - Alonso del Arte, Dec 06 2011
Also 2*sqrt(2) is the ratio of the perimeter of a square to its diameter (diagonal length). - Rick L. Shepherd, Dec 29 2016
Uchiyama shows that every interval (n, n + c*n^(1/4)) contains an integer that is the sum of two squares, where c = 2^(3/2). - Michel Marcus, Jan 03 2018
This is the area of the eighth-smallest triangle with integer side lengths (2, 3, 3), or the seventh-smallest triangle if two smaller triangles with the same area are counted only once (see A331251). - Hugo Pfoertner, Feb 12 2020
Diameter of a sphere whose surface area equals 8*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Feb 13 2020
Sqrt(8) = area between the curves y = sin(x) and y = cos(x) for Pi/4 < x < 5 Pi/4; this is one of infinitely many congruent convex regions bounded by the two curves. - Clark Kimberling, May 03 2020
Area of the regular 8-gon with circumradius =1. - R. J. Mathar, Aug 24 2023

			2.828427124746190097603377448419396157139343750753896146353359475981464...
Sqrt(8) = sqrt(1+2*i*sqrt(2)) + sqrt(1-2*i*sqrt(2)) = sqrt(1/2+2*i*sqrt(3)) + sqrt(1/2-2*i*sqrt(3)), where i=sqrt(-1). - _Bruno Berselli_, Nov 20 2012
1 + 3/4 + 3*5/(4*8) + 3*5*7/(4*8*12) + 3*5*7*9/(4*8*12*16) + ... - _Bruno Berselli_, Mar 16 2014

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 187.
S. R. Finch, Moving Sofa Constant, Sect. 8.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 519-523, 2003.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Jason Kimberley, Index of expansions of sqrt(d) in base b
R. J. Nemiroff & J. Bonnell, The first 1 million digits of the square root of 8
R. J. Nemiroff & J. Bonnell, Plouffe's Inverter, The first 1 million digits of the square root of 8
Ana Rechtman, Juin 2023, 3e défi, Images des Mathématiques, CNRS, 2023.
S. Uchiyama, On the distribution of integers representable as a sum of two h-th powers, J. Fac. Sci. Hokkaido Univ. Ser. I, 18, 124-127, 1964/1965.
Eric Weisstein's World of Mathematics, Moving Ladder Problem
Index entries for algebraic numbers, degree 2

Cf. A040005 (continued fraction).

Cf. A331250, A331251.

SetDefaultRealField(RealField(100)); Sqrt(8); // Vincenzo Librandi, Feb 13 2020

evalf(2^(3/2)) ; # R. J. Mathar, Jul 15 2013

RealDigits[N[Sqrt[8],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

default(realprecision, 20080); x=sqrt(8); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010466.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

Equals 1 + Sum_{n>=1} ( Product_{k=1..n} (2k+1)/(4k) ). - Bruno Berselli, Mar 16 2014
Equals 2*A002193. - R. J. Mathar, Jan 14 2021
From Peter Bala, Mar 01 2022: (Start)
Equals 3*Sum_{n >= 0} (1/(4*n+1) - 1/(4*n-3))*binomial(1/2,n). Cf. A002580 and A175576.
Equals 4*hypergeom([-1/2, -3/4], [5/4], -1). (End)
Equals 8 * A020765. - R. J. Mathar, Aug 24 2023

A040001 1 followed by {1, 2} repeated.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane

Continued fraction for sqrt(3).
Also coefficient of the highest power of q in the expansion of the polynomial nu(n) defined by: nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,1), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
nu(0)=1 nu(1)=1; nu(2)=2; nu(3)=3+q; nu(4)=5+3q+2q^2; nu(5)=8+7q+6q^2+4q^3+q^4; nu(6)=13+15q+16q^2+14q^3+11q^4+5q^5+2q^6.
From Jaroslav Krizek, May 28 2010: (Start)
a(n) = denominators of arithmetic means of the first n positive integers for n >= 1.
See A026741(n+1) or A145051(n) - denominators of arithmetic means of the first n positive integers. (End)
From R. J. Mathar, Feb 16 2011: (Start)
This is a prototype of multiplicative sequences defined by a(p^e)=1 for odd primes p, and a(2^e)=c with some constant c, here c=2. They have Dirichlet generating functions (1+(c-1)/2^s)*zeta(s).
Examples are A153284, A176040 (c=3), A040005 (c=4), A021070, A176260 (c=5), A040011, A176355 (c=6), A176415 (c=7), A040019, A021059 (c=8), A040029 (c=10), A040041 (c=12). (End)
a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A000325(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
For n > 0: denominators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n; see A226555 for numerators. - Reinhard Zumkeller, Jun 10 2013
From Jianing Song, Nov 01 2022: (Start)
For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.
The maximal gap is given by A048669. (End)

			1.732050807568877293527446341... = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...))))
G.f. = 1 + x + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + ...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
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
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Ashok Kumar Gupta and Ashok Kumar Mittal, Bifurcating continued fractions, arXiv:math/0002227 [math.GM] (2000).
Michael Somos, Rational Function Multiplicative Coefficients.
Eric Weisstein's World of Mathematics, Square Root.
Eric Weisstein's World of Mathematics, Theodorus's Constant.
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000034, A002194, A133566, A083329 (binomial Transf).

Apart from a(0) the same as A134451.

a040001 0 = 1; a040001 n = 2 - mod n 2
a040001_list = 1 : cycle [1, 2]  -- Reinhard Zumkeller, Apr 16 2015

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

ContinuedFraction[Sqrt[3],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PadRight[{1},120,{2,1}] (* Harvey P. Dale, Nov 26 2015 *)

{a(n) = 2 - (n==0) - (n%2)} /* Michael Somos, Jun 11 2003 */

{ allocatemem(932245000); default(realprecision, 12000); x=contfrac(sqrt(3)); for (n=0, 20000, write("b040001.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

Multiplicative with a(p^e) = 2 if p even; 1 if p odd. - David W. Wilson, Aug 01 2001
G.f.: (1 + x + x^2) / (1 - x^2). E.g.f.: (3*exp(x)-2*exp(0)+exp(-x))/2. - Paul Barry, Apr 27 2003
a(n) = (3-2*0^n +(-1)^n)/2. a(-n)=a(n). a(2n+1)=1, a(2n)=2, n nonzero.
a(n) = sum{k=0..n, F(n-k+1)*(-2+(1+(-1)^k)/2+C(2, k)+0^k)}. - Paul Barry, Jun 22 2007
Row sums of triangle A133566. - Gary W. Adamson, Sep 16 2007
Euler transform of length 3 sequence [ 1, 1, -1]. - Michael Somos, Aug 04 2009
Moebius transform is length 2 sequence [ 1, 1]. - Michael Somos, Aug 04 2009
a(n) = sign(n) + ((n+1) mod 2) = 1 + sign(n) - (n mod 2). - Wesley Ivan Hurt, Dec 13 2013

A187110 Decimal expansion of sqrt(3/8).

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Apart from leading digits, the same as A174925.

Radius of the circumscribed sphere (congruent with vertices) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			sqrt(3/8)=0.61237243569579452454932101867647284799148687016417..

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Michael Penn, Maximize the area of one ellipse!, YouTube video, 2020.
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010464, A010466, A020781, A040001, A040005, A115754, A301755.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron). - Stanislav Sykora, Feb 10 2014

Mathematica
```
RealDigits[N[Sqrt[3/8],300]][[1]]
```

sqrt(3/8) \\ Charles R Greathouse IV, Mar 16 2022

Equals A010464/4. - Stefano Spezia, Jan 26 2025

Equals 3*A020781 = A115754/2 = sqrt(A301755). - Hugo Pfoertner, Jan 26 2025

A041010 Numerators of continued fraction convergents to sqrt(8).

Original entry on oeis.org

2, 3, 14, 17, 82, 99, 478, 577, 2786, 3363, 16238, 19601, 94642, 114243, 551614, 665857, 3215042, 3880899, 18738638, 22619537, 109216786, 131836323, 636562078, 768398401, 3710155682, 4478554083, 21624372014, 26102926097, 126036076402, 152139002499

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [1 removed by _Georg Fischer_, Jul 01 2019]
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A040005 (continued fraction), A041011 (denominators), A010466 (decimals).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[8],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
CoefficientList[Series[(2 + 3*x + 2*x^2 - x^3)/(1 - 6*x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify
a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

A041010=contfracpnqn(c=contfrac(sqrt(8)),#c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041010[n+1]! For more terms use:
A041010(n)={n<#A041010|| A041010=extend(A041010, [-1,0,6,0]~, n\.8); A041010[n+1]}
extend(A,c,N)={for(n=#A+1,#A=Vec(A,N), A[n]=A[n-#c..n-1]*c);A} \\ (End)

a(n) = 6*a(n-2) - a(n-4).
a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).
a(2n) = A001333(2n), a(2n+1) = 2*A001333(2n+1).
G.f.: (2+3*x+2*x^2-x^3)/(1-6*x^2+x^4).
a(n) = A001333(n+1)*A000034(n+1). - R. J. Mathar, Jul 08 2009
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2))) + (-1+sqrt(2))*(3+2*sqrt(2))^n.
a1(n) = ((3-2*sqrt(2))^n + (3+2*sqrt(2))^n)/2. (End)

Entry improved by Michael Somos
Initial term 1 removed and b-file, program and formulas adapted by Georg Fischer, Jul 01 2019
Cross-references added by M. F. Hasler, Nov 02 2019

A013648 Numbers k such that the periodic part of the continued fraction for sqrt(k) contains a single 1.

Original entry on oeis.org

3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 175, 176, 195, 208, 224, 255, 288, 323, 360, 399, 440, 483, 528, 551, 575, 624, 675, 728, 783, 799, 840, 899, 960, 1023, 1035, 1088, 1155, 1224, 1247, 1295, 1368, 1403, 1443, 1520, 1599, 1680, 1763, 1848, 1872

Offset: 1

Clark Kimberling

nonn

All the terms of A005563 are here, as well as some additional terms (with even period > 2 and the digit 1 in central position) (e.g., sqrt(175) = [13,'4, 2, 1, 2, 4, 26']).

Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Robert Macmillan, Continued fractions, Math. Gaz., Vol. 84, No. 499 (2000), pp. 30-35. See p. 34.

Union of A005563 and A102538.

Cf. A040001, A040005, A040011, A040019, A040029, A062196.

Select[ Range@ 1900, !IntegerQ[ Sqrt@ #] && Count[ ContinuedFraction[ Sqrt@ #][[2]], 1] == 1 &] (* Robert G. Wilson v, Jul 03 2011 *)

Additional comments from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 30 2001

Incorrect formulas and programs removed by R. J. Mathar, Jan 06 2011

A040033 Continued fraction for sqrt(40).

Original entry on oeis.org

6, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3

Offset: 0

N. J. A. Sloane

			6.3245553203367586639977870... = 6 + 1/(3 + 1/(12 + 1/(3 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 05 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
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010494 (decimal expansion), A040005.

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

ContinuedFraction[Sqrt[40],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
LinearRecurrence[{0,1},{6,3,12},100] (* or *) PadRight[{6},100,{12,3}] (* Harvey P. Dale, Mar 03 2018 *)

{ allocatemem(932245000); default(realprecision, 32000); x=contfrac(sqrt(40)); for (n=0, 20000, write("b040033.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 05 2009

From Stefano Spezia, Jul 27 2025: (Start)
a(n) = 3*A040005(n).
G.f.: 3*(2 + x + 2*x^2)/(1 - x^2). (End)

A109054 Squares and numbers k such that the continued fraction expansion of sqrt(k) is multiplicative.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 9, 13, 14, 15, 16, 22, 23, 24, 25, 32, 33, 34, 35, 36, 44, 47, 48, 49, 58, 59, 60, 62, 63, 64, 74, 75, 78, 79, 80, 81, 95, 96, 98, 99, 100, 114, 119, 120, 121, 135, 136, 138, 140, 141, 142, 143, 144, 160, 162, 164, 167, 168, 169, 185, 187, 189, 192

Offset: 1

Mitch Harris, Jun 18 2005

If we consider each square k as having a continued fraction expansion c of all zeros after c(0) = sqrt(k)-1, then the continued fraction expansion of sqrt(k) for each square is trivially multiplicative.

For nonsquares, c(1) must be 1 and so k must satisfy m + 1/2 < sqrt(k) <= m+1, for some integer m.

			The continued fraction of sqrt(22) is c = (4; 1, 2, 4, 2, 1, 8, ...) = A010126, which is multiplicative with c(2^e) = 2, c(3^e) = 4, c(p^e) = 1 otherwise.

Union of A000290 and A108575.

Continued fraction expansions: A040001, A010121, A040005, etc.

A190567 Continued fraction expansion of 46*sqrt(46).

Original entry on oeis.org

311, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622

Offset: 0

Bruno Berselli, May 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010136; A040005, A040021, A010186, A040201, A040324, A040489, A040968.

[311] cat &cat[ [1,76,1,622]: n in [1..18] ];

I:=[311,1,76,1,622]; [n le 5 select I[n] else Self(n-4): n in [1..80]]; // Vincenzo Librandi, Jun 14 2013

ContinuedFraction[46 Sqrt[46], 80] (* or *) PadRight[{311}, 80, {622, 1, 76, 1}]
CoefficientList[Series[(311 + x + 76 x^2 + x^3 + 311 x^4) / (1 - x^4), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 14 2013 *)

a(n)=if(n,[622,1,76,1][n%4+1],311) \\ Charles R Greathouse IV, May 13 2011

G.f.:  (311+x+76*x^2+x^3+311*x^4)/(1-x^4).
a(n) = 1+3*(1+(-1)^n)*(116+91*i^n)/2 with n>0, i=sqrt(-1) and a(0)=311.
a(n) = (-1513*(n mod 4)+575*((n+1) mod 4)+125*((n+2) mod 4)+2213*((n+3) mod 4))/12  for n>0.
a(n) = a(n-4), n>=5. - Vincenzo Librandi, Jun 14 2013

cp's OEIS Frontend

A010466 Decimal expansion of square root of 8.

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A040001 1 followed by {1, 2} repeated.

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A187110 Decimal expansion of sqrt(3/8).

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A041010 Numerators of continued fraction convergents to sqrt(8).

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A013648 Numbers k such that the periodic part of the continued fraction for sqrt(k) contains a single 1.

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A040033 Continued fraction for sqrt(40).

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