A010465 -id:A010465 - cp's OEIS frontend

A248237 Egyptian fraction representation of sqrt(7) (A010465) using a greedy function.

2, 2, 7, 346, 250326, 159992246122, 43126926376468440463866, 2067900185855597116733968004943580535040713497, 14833490144163739987168640921306687956266487136609932761918465200939453258507455567518894133

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[ 7]]

A010121 Continued fraction for sqrt(7).

Original entry on oeis.org

2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4

Offset: 0

N. J. A. Sloane

This is a basic member of a family of 4-periodic multiplicative sequences with two parameters (c1,c2), defined for n >= 1 by a(n)=1 if n is odd, a(n)=c1 if n == 0 (mod 4) and a(n)=c2 if n == 2 (mod 4). Here, (c1,c2)=(4,1).
The Dirichlet generating function is (1+(c2-1)/2^s+(c1-c2)/4^s)*zeta(s).
Other members are A010123 with parameters (6,2), A010127 (8,3), A010130 (10,1), A010131 (10,2), A010132 (10,4), A010137 (12,5), A010146 (14,6), A089146 (4,8), A109008 (4,2), A112132 (7,3). If c1=c2, this reduces to the cases discussed in A040001. - R. J. Mathar, Feb 18 2011

			2.645751311064590590501615753...  = A010465 = 2 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...)))).

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
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4, example 5.
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010465 (decimal expansion).

ContinuedFraction[Sqrt[7],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
CoefficientList[Series[(2 x^2 + 3 x + 2) (x^2 - x + 1) / ((1 - x) (1 + x) (x^2 + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 26 2016 *)
PadRight[{2},120,{4,1,1,1}] (* Harvey P. Dale, Nov 30 2019 *)

{ allocatemem(932245000); default(realprecision, 13000); x=contfrac(sqrt(7)); for (n=0, 20000, write("b010121.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

From R. J. Mathar, Jun 17 2009: (Start)
G.f.: -(2*x^2+3*x+2)*(x^2-x+1)/((x-1)*(1+x)*(x^2+1)).
a(n) = a(n-4), n > 4. (End)
a(n) = (7 + 3*(-1)^n + 3*(-i)^n + 3*i^n)/4, n > 0, where i is the imaginary unit. - Bruno Berselli, Feb 18 2011

A041008 Numerators of continued fraction convergents to sqrt(7).

Original entry on oeis.org

2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257, 149858, 182115, 331973, 514088, 2388325, 2902413, 5290738, 8193151, 38063342, 46256493, 84319835, 130576328, 606625147, 737201475, 1343826622, 2081028097, 9667939010, 11748967107, 21416906117

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4.
C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,16,0,0,0,-1).

Cf. A010465, A041009 (denominators), A266698 (quadrisection), A001081 (quadrisection).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041010 (m=8), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[7],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
Numerator[Convergents[Sqrt[7], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
LinearRecurrence[{0,0,0,16,0,0,0,-1},{2,3,5,8,37,45,82,127},40] (* Harvey P. Dale, Jul 23 2021 *)

A041008=contfracpnqn(c=contfrac(sqrt(7)),#c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041008[n+1]! For more terms use:
A041008(n)={n<#A041008|| A041008=extend(A041008, [4, 16; 8, -1], n\.8); A041008[n+1]}
extend(A,c,N)={for(n=#A+1, #A=Vec(A, N), A[n]=[A[n-i]|i<-c[,1]]*c[,2]); A} \\ (End)

G.f.: (2 + 3*x + 5*x^2 + 8*x^3 + 5*x^4 - 3*x^5 + 2*x^6 - x^7)/(1 - 16*x^4 + x^8).

A154244 a(n) = 6a(n-1) - 2a(n-2) for n>1; a(1)=1, a(2)=6.

Original entry on oeis.org

1, 6, 34, 192, 1084, 6120, 34552, 195072, 1101328, 6217824, 35104288, 198190080, 1118931904, 6317211264, 35665403776, 201358000128, 1136817193216, 6418187159040, 36235488567808, 204576557088768, 1154988365396992

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Binomial transform of A126473.
lim_{n -> infinity} a(n)/a(n-1) = 3+sqrt(7) = 5.6457513110....
a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} avoiding 01 and 02. - Milan Janjic, Dec 17 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. See Cor. 3.7(e).
Index entries for linear recurrences with constant coefficients, signature (6,-2).

Equals 1 followed by 2*A010913 (Pisot sequence E(3,17)).

Cf. A010465 (decimal expansion of square root of 7), A126473.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((3+r)^n-(3-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

I:=[1, 6]; [n le 2 select I[n] else 6*Self(n-1)-2*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 02 2012

a[n_]:=(MatrixPower[{{1,3},{1,5}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{6, -2}, {1, 6}, 40] (* Vincenzo Librandi, Feb 02 2012 *)

a[1]:1$ a[2]:6$ a[n]:=6*a[n-1]-2*a[n-2]$ makelist(a[n], n, 1, 21); /* Bruno Berselli, May 30 2011 */

Vec(1/(1-6*x+2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Dec 28 2011

[lucas_number1(n,6,2) for n in range(1, 22)] # Zerinvary Lajos, Apr 22 2009

a(n) = ((3 + sqrt(7))^n - (3 - sqrt(7))^n)/(2*sqrt(7)).

G.f.: x/(1-6*x+2*x^2). - Philippe Deléham, Jan 06 2009

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009
Edited by Klaus Brockhaus, Oct 06 2009
Name (corrected) from Philippe Deléham, Jan 06 2009

A233587 Coefficients of the generalized continued fraction expansion sqrt(7) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

Original entry on oeis.org

2, 3, 30, 34, 111, 235, 3775, 5052, 7352, 9091, 34991, 35530, 53424, 57290, 66023, 1409179, 1519111, 1725990, 1812396, 4370835, 4507156, 4655396, 44257080, 234755198, 261519946, 264374278, 273487975

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details on Blazys' expansions, see A233582.

Sqrt(7) is the first square root of a natural number with an a-periodic Blazys' expansion (see A233592 and A233593).

Stanislav Sykora, Table of n, a(n) for n = 1..1000

Cf. Blazys' expansions: A233582 (Pi), A233583 (e), A233584 (sqrt(e)), A233585 (1/gamma), A233585 (2*gamma) and Blazys' continued fractions: A233588, A233589, A233590, A233591.

Cf. A010465, A010121.

BlazysExpansion[n_, mx_] := Block[{k = 1, x = n, lmt = mx + 1, s, lst = {}}, While[k < lmt, s = Floor[x]; x = 1/(x/s - 1); AppendTo[lst, s]; k++]; lst]; BlazysExpansion[Sqrt@7, 32] (* Robert G. Wilson v, May 22 2014 *)

bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
bx(sqrt(7), 1000) \\ Execution; use very high real precision

sqrt(7) = 2+2/(3+3/(30+30/(34+34/(111+...)))).

A114765 a(n) = floor(sqrt(7) * 10^n)^2.

Original entry on oeis.org

4, 676, 69696, 6996025, 699972849, 69999930625, 6999998354001, 699999994145169, 69999999943667161, 6999999999658218721, 699999999965821872100, 69999999999757088783236, 6999999999996874888812096, 699999999999952064012316025, 69999999999999968753591518681

Offset: 0

Amarnath Murthy, Nov 17 2005

nonn

Largest square less than 7 * 10^(2n).

			sqrt(7) = 2.645751311...
floor(sqrt(7) * 10) = 26 and 26^2 = 676, so a(1) = 676.
floor(sqrt(7) * 100) = 264 and 264^2 = 69696, so a(2) = 69696.
floor(sqrt(7) * 1000) = 2645 and 2645^2 = 6996025, so a(3) = 6996025.

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A114761, A114762, A114763, A114764.

Cf. A010465 (sqrt(7)).

[Floor(7^(1/2)*10^n)^2: n in [0..150]]; // Vincenzo Librandi, Feb 05 2011

$MaxExtraPrecision := 200; Table[Floor[7^(1/2) * 10^n]^2, {n, 0, 20}] (* Stefan Steinerberger, Jan 26 2006 *)

a(n)={sqrtint(7*10^(2*n))^2} \\ Andrew Howroyd, Nov 09 2019

More terms from Stefan Steinerberger, Jan 26 2006

Terms a(12) and beyond from Andrew Howroyd, Nov 09 2019

A041009 Denominators of continued fraction convergents to sqrt(7).

Original entry on oeis.org

1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192, 56641, 68833, 125474, 194307, 902702, 1097009, 1999711, 3096720, 14386591, 17483311, 31869902, 49353213, 229282754, 278635967

Offset: 0

N. J. A. Sloane

Sqrt(7) = 2 + 9/14 + 9/(14*223) + 9/(223*3554) + 9/(3554*56641) + ...; sum of these 5 terms = 2.64575131088, with sqrt(7) = 2.64575131106... The terms 14, 223, 3554, ... = a(4), a(8), a(12), ... - Gary W. Adamson, Dec 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6
Index entries for linear recurrences with constant coefficients, signature (0,0,0,16,0,0,0,-1).

Cf. A010465, A041008.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[7],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
Denominator[Convergents[Sqrt[7],30]] (* or *) LinearRecurrence[ {0,0,0,16,0,0,0,-1},{1,1,2,3,14,17,31,48},30] (* Harvey P. Dale, Dec 17 2019 *)

G.f.: (1+x+2*x^2+3*x^3-2*x^4+x^5-x^6)/(1-16*x^4+x^8). - Colin Barker, Mar 13 2012

A010483 Decimal expansion of square root of 28.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 5 followed by {3, 2, 3, 10} repeated. - Harry J. Smith, Jun 04 2009

			5.2915026221291811810032315072785208514205183661649003607366689184... - _Harry J. Smith_, Jun 04 2009

Harry J. Smith, Table of n, a(n) for n = 1..20000
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Index entries for algebraic numbers, degree 2.

Cf. A040022 Continued fraction. - Harry J. Smith, Jun 04 2009

RealDigits[N[Sqrt[28], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

{ default(realprecision, 20080); x=sqrt(28); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010483.txt", n, " ", d)); } \\ Harry J. Smith, Jun 04 2009

Equals 2*A010465. - R. J. Mathar, Jan 14 2021

A020764 Decimal expansion of 1/sqrt(7).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

1/sqrt(7) = 0.377964473009227227214516536234180060815751311868921454338333494171... - Vladimir Joseph Stephan Orlovsky, May 27 2010

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 2.

Cf. A010465.

RealDigits[N[1/Sqrt[7],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)
realDigitsRecip[Sqrt[7]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Sep 29 2024 *)

Equals 1/A010465.

A004575 Expansion of sqrt(7) in base 8.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010465.

Prune(Reverse(IntegerToSequence(Isqrt(7*8^204),8))); // Jason Kimberley, Feb 19 2012

RealDigits[Sqrt[7],8,120][[1]] (* Harvey P. Dale, Jan 04 2018 *)

More terms from Jason Kimberley, Feb 27 2012

cp's OEIS Frontend

A248237 Egyptian fraction representation of sqrt(7) (A010465) using a greedy function.

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A010121 Continued fraction for sqrt(7).

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A041008 Numerators of continued fraction convergents to sqrt(7).

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A154244 a(n) = 6*a(n-1) - 2*a(n-2) for n>1; a(1)=1, a(2)=6.

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A233587 Coefficients of the generalized continued fraction expansion sqrt(7) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

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A114765 a(n) = floor(sqrt(7) * 10^n)^2.

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A041009 Denominators of continued fraction convergents to sqrt(7).

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A154244 a(n) = 6a(n-1) - 2a(n-2) for n>1; a(1)=1, a(2)=6.