A090654 -id:A090654 - cp's OEIS frontend

A090388 Decimal expansion of 1 + sqrt(3).

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

Offset: 1

Felix Tubiana, Feb 05 2004

1 + sqrt(3) is the length of the minimal Steiner network that connects the four vertices of a unit square. - Lekraj Beedassy, May 02 2008
This is the case n = 12 in the identity (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)) = 1 + 2*cos(2*Pi/n). - Bruno Berselli, Dec 14 2012
Equals n + n/(n + n/(n + n/(n + ...))) for n = 2. - Stanislav Sykora, Jan 23 2014
A non-optimal solution to the problem of finding the length of shortest fence that protects privacy of a square garden [Kawohl]. Cf. A256965. - N. J. A. Sloane, Apr 14 2015
Perimeter of a 30-60-90 triangle with longest leg equal to 1. - Wesley Ivan Hurt, Apr 09 2016
Length of the second shortest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Dec 13 2020
Surface area of a square pyramid (Johnson solid J_1) with unit edges. - Paolo Xausa, Aug 04 2025

			2.7320508075688772...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Bernd Kawohl, Some nonconvex shape optimization problems, in: Optimal Shape Design, Eds. A.Cellina u. A. Ornelas, Springer Lecture Notes in Math.1740 (2000), p. 7-46.
Ian Stewart, Pursuing Polygonal Privacy, Mathematical Recreations Column, Scientific American, 284 (No. 2, 2001), 88-89.
Wikipedia, Square pyramid.
Index entries for algebraic numbers, degree 2

Cf. n + n/(n + n/(n + ...)): A090458 (n = 3), A090488 (n = 4), A090550 (n = 5), A092294 (n = 6), A092290 (n = 7), A090654 (n = 8), A090655 (n = 9), A090656 (n = 10). - Stanislav Sykora, Jan 23 2014

Cf., also A256965.

Digits:=100: evalf((1+sqrt(3))); # Wesley Ivan Hurt, Apr 09 2016

RealDigits[1 + Sqrt[3], 10, 100][[1]] (* Alonso del Arte, Feb 23 2014 *)

1 + sqrt(3) \\ Michel Marcus, Apr 10 2016

Equals 1 + A002194. - R. J. Mathar, Oct 16 2015

Equals A019973 -1 . - R. J. Mathar, May 25 2023

Better definition from Rick L. Shepherd, Jul 02 2004

A090488 Decimal expansion of 2 + 2*sqrt(2).

Original entry on oeis.org

4, 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

Offset: 1

Felix Tubiana, Feb 05 2004

Side length of smallest square containing five circles of radius 1. - Charles R Greathouse IV, Apr 05 2011
Equals n + n/(n +n/(n +n/(n +....))) for n = 4.  See also A090388. - Stanislav Sykora, Jan 23 2014
Also the area of a regular octagon with unit edge length. - Stanislav Sykora, Apr 12 2015
The positive solution to x^2 - 4*x - 4 = 0. The negative solution is -1 * A163960 = -0.82842... . - Michal Paulovic, Dec 12 2023

			4.828427124746190097603377448419396157139343750...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Octagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 2

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Cf. A002193, A010466, A014176, A086178, A156035, A157258, A163960, A365823.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A256853, A178816, A256854, A178809.

RealDigits[2+2Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 11 2015 *)

2*(1 + sqrt(2)) \\ G. C. Greubel, Jul 03 2017

Equals 1 + A086178 = 2*A014176. - R. J. Mathar, Sep 03 2007
From Michal Paulovic, Dec 12 2023: (Start)
Equals A010466 + 2.
Equals A156035 - 1.
Equals A157258 - 5.
Equals A163960 + 4.
Equals A365823 - 2.
Equals [4; 1, 4, ...] (periodic continued fraction expansion).
Equals sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * ...)))). (End)

Better definition from Rick L. Shepherd, Jul 02 2004

A090550 Decimal expansion of solution to n/x = x - n for n = 5.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

n/x = x - n with n = 1 gives the Golden Ratio = 1.6180339887...

Equals n + n/(n + n/(n + n/(n + ....))) for n = 5. See also A090388. - Stanislav Sykora, Jan 23 2014

			5.85410196624968454...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Index entries for algebraic numbers, degree 2.

Cf. n + n/(n + n/(n + ...)): A090388 (n = 2), A090458 (n = 3), A090488 (n = 4), A092294 (n = 6), A092290 (n = 7), A090654 (n = 8), A090655 (n = 9), A090656 (n = 10). - Stanislav Sykora, Jan 23 2014

Cf. A001622, A057088.

RealDigits[(5 + 3 Sqrt[5])/2, 10, 120][[1]] (* Harvey P. Dale, Nov 27 2013 *)

(5 + 3*sqrt(5))/2 \\ G. C. Greubel, Jul 03 2017

n/x = x - n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 5: x = (5 + sqrt(45))/2 = 5.85410196624968454...
Equals (5 + 3*sqrt(5))/2 = 1 + 3*phi = sqrt(5)*(phi)^2, where phi is the golden ratio. - G. C. Greubel, Jul 03 2017
Equals 2*phi^3 - phi^2. - Michel Marcus, Apr 20 2020
Minimal polynomial is x^2 - 5x - 5 (this number is an algebraic integer). - Alonso del Arte, Apr 20 2020(n).
Equals lim_{n->oo} A057088(n+1)/A057088(n) = 1 + 3*phi. - Wolfdieter Lang, Nov 16 2023

A090458 Decimal expansion of (3 + sqrt(21))/2.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Decimal expansion of the solution to n/x = x-n for n-3. n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...
n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 3: x = (3 + sqrt(21))/2 = 3.79128784747792...
x=3.7912878474... is the shape of a rectangle whose geometric partition (as at A188635) consists of 3 squares, then 1 square, then 3 squares, etc., matching the continued fraction of x, which is [3,1,3,1,3,1,3,1,3,1,...]. (See the Mathematica program below.) - Clark Kimberling, May 05 2011
x appears to be the limit for n to infinity of the ratio of the number of even numbers that take n steps to reach 1 to the number of odd numbers that take n steps to reach 1 in the Collatz iteration. As A005186(n-1) is the number of even numbers that take n steps to reach 1, this means x = lim A005186(n-1)/A176866(n). - Markus Sigg, Oct 20 2020
From Wolfdieter Lang, Sep 02 2022: (Start)
This integer in the quadratic number field Q(sqrt(21)) equals the (real) cube root of 27 + 6*sqrt(21) = 54.4954541... . See Euler, Elements of Algebra, Article 748 or Algebra (in German) p. 306, Kapitel 12, 187.
Subtracting 3 from the present number gives the (real) cube root of
  -27 +  6*sqrt(21) = 0.4954541... . (End)

			3.79128784747792...

Leonhard Euler, Vollständige Anleitung zur Algebra, (1770), Reclam, Leipzig, 1883, p.306, Kapitel 12, 187.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Leonhard Euler, Elements of Algebra, p. 244, Article 748.
Markus Sigg, Heuristic argument for the asymptotic value of the even/odd ratio of number of steps in the Collatz iteration, 2020.
Index entries for algebraic numbers, degree 2

Of the same type as this: A090388 (n=2), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10).
Equals 3*A176014 (constant).
Cf. A356034.

FromContinuedFraction[{3,1,{3,1}}]
ContinuedFraction[%,20]
RealDigits[N[%%,120]] (*A090458*)
N[%%%,40]
RealDigits[(3 + Sqrt[21])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

solve(x=3,4,x^2-3*x-3) \\ Charles R Greathouse IV, Oct 04 2011

(3+sqrt(21))/2 \\ Charles R Greathouse IV, Oct 04 2011

Equals (27 + 6*sqrt(21))^(1/3). - Wolfdieter Lang, Sep 01 2022

Additional comments from Rick L. Shepherd, Jul 02 2004

A010480 Decimal expansion of square root of 24.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 4 followed by {1, 8} repeated. - Harry J. Smith, Jun 03 2009
The solution to the Lane-Emden equation for a sphere of polytropic index n = 0. - Arkadiusz Wesolowski, Nov 11 2014
Using Bretschneider's formula this is the maximum area of a quadrilateral with side lengths of 1,2,3 and 4. - Scott R. Shannon, Jan 06 2020

			4.898979485566356196394568149411782783931894961313340256865385134501920....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Wikipedia, Bretschneider's formula.
Index entries for algebraic numbers, degree 2

Cf. A040019 (continued fraction). - Harry J. Smith, Jun 03 2009

Cf. A010464, A090654.

RealDigits[N[Sqrt[24],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(24); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010480.txt", n, " ", d));  \\ Harry J. Smith, Jun 03 2009

Equals 2*A010464. - R. J. Mathar, Jan 14 2021

Equals 4 + Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1) * 8^k). - Amiram Eldar, May 06 2022

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

A092290 Decimal expansion of solution to n/x = x-n for n = 7.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...

Equals n +n/(n +n/(n +n/(n +....))) for n = 7. See also A090388. - Stanislav Sykora, Jan 23 2014

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Index entries for algebraic numbers, degree 2.

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014

RealDigits[(7+Sqrt[77])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

(7+sqrt(77))/2 \\ Charles R Greathouse IV, Oct 18 2012

n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 7: x = (7 + sqrt(77))/2 = 7.88748219369606...

A092294 Decimal expansion of 3 + sqrt(15).

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Equals n +n/(n +n/(n +n/(n +....))) for n = 6. See also A090388. - Stanislav Sykora, Jan 23 2014

			6.87298334620741688...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014

RealDigits[3+Sqrt[15],10,120][[1]] (* Harvey P. Dale, Aug 29 2012 *)

sqrt(15)+3 \\ Charles R Greathouse IV, Apr 25 2016

Equals A010472 plus 3. - R. J. Mathar, Sep 08 2008

Equals 1/A176016 + 6. - Hugo Pfoertner, Mar 19 2024

A090655 Decimal expansion of solution to n/x = x-n for n = 9.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...

Equals n +n/(n +n/(n +n/(n +....))) for n = 9. See also A090388. - Stanislav Sykora, Jan 23 2014

			9.90832691319598...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090656 (n=10). - Stanislav Sykora, Jan 23 2014

RealDigits[(3/2)*(3+Sqrt[13]), 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

(3/2)*(3 + sqrt(13)) \\ G. C. Greubel, Jul 03 2017

n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 9: x = (9 + sqrt(117))/2 = 9.90832691319598...

Equals (3/2)*(3 + sqrt(13)). - G. C. Greubel, Jul 03 2017

A090656 Decimal expansion of 5 + sqrt(35).

Original entry on oeis.org

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

Offset: 2

Felix Tubiana, Feb 05 2004

Equals n+n/(n+n/(n+n/(n+....))) for n = 10. See also A090388. - Stanislav Sykora, Jan 23 2014

			10.9160797830996160...

Chai Wah Wu, Table of n, a(n) for n = 2..10003
Index entries for algebraic numbers, degree 2

Equals A010490 plus 5. - R. J. Mathar, Sep 08 2008
Cf. A161321.
Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9). - Stanislav Sykora, Jan 23 2014

RealDigits[5+Sqrt[35],10,120][[1]] (* Harvey P. Dale, Jun 17 2014 *)

sqrt(35)+5 \\ Charles R Greathouse IV, Apr 25 2016

A200728 Decimal expansion of the circumradius of cyclic quadrilateral with sides 1, 2, 3, 4.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Nov 21 2011

Area is K = sqrt(24) = 4.89897948556635619639... (decimal expansion in A010480).

			2.0026024734496526299517...

Zak Seidov, Figure showing polygon.
Index entries for algebraic numbers, degree 2

Cf. A090654, A010480, A200113, A200257.

RealDigits[Sqrt[385/96], 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)

sqrt(385/96) \\ Charles R Greathouse IV, Dec 28 2011

R = (385/96)^(1/2).

cp's OEIS Frontend

A090388 Decimal expansion of 1 + sqrt(3).

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A090488 Decimal expansion of 2 + 2*sqrt(2).

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A090550 Decimal expansion of solution to n/x = x - n for n = 5.

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A090458 Decimal expansion of (3 + sqrt(21))/2.

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A010480 Decimal expansion of square root of 24.

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A092290 Decimal expansion of solution to n/x = x-n for n = 7.

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A092294 Decimal expansion of 3 + sqrt(15).