A090458 -id:A090458 - 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

A090654 Decimal expansion of 4 + 2*sqrt(6).

Original entry on oeis.org

8, 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

Offset: 1

Felix Tubiana, Feb 05 2004

Equals n +n/(n +n/(n +n/(n +....))) for n = 8. See also A090388. - Stanislav Sykora, Jan 23 2014

			8.898979485566356196394568149...

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

Cf. A010480, A176213.
Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Essentially the same as A010480.

RealDigits[4 + 2*Sqrt[6], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

4 + 2*sqrt(6) \\ G. C. Greubel, Jul 03 2017

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

A222134 Decimal expansion of sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))).

Original entry on oeis.org

2, 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

Jaroslav Krizek, Feb 08 2013

c^n = A015440(n) + A015440(n-1) * A222135, where c = (1 + sqrt(21))/2 and A222135 = (-1 + sqrt(21))/2. - Gary W. Adamson, Nov 27 2023

			2.791287847477920003294023596864...

Index entries for algebraic numbers, degree 2.

Cf. A090458, A107905, A222135, A015440.

Mathematica
```
RealDigits[(1 + Sqrt[21])/2, 10, 130]
```

(sqrt(21) + 1)/2 \\ Charles R Greathouse IV, Feb 07 2025

Equals (sqrt(21) + 1)/2 = A090458 - 1 = A107905 - 2 = A222135 + 1.
sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))) - 1 = sqrt(5 - sqrt(5 - sqrt(5 - sqrt(5 - ... )))) = A222135.
Minimal polynomial: x^2 - x - 5. - Stefano Spezia, Jul 02 2025

A107905 Decimal expansion of (5+sqrt(21))/2.

Original entry on oeis.org

4, 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

Offset: 1

Jonathan Vos Post, Jun 22 2007

			4.7912878474779200032940235968640042444922282883839859513036...
The zeros at 15, 16 and 17 digits after the decimal point allow for a good rational approximation. The continued fraction is [4,1,3,1,3,1,3,...] = 4 + 1/(1+ 1/(3+ 1/(1+ 1/(3+ 1/(1+ 1/(3+ 1(/1+ ...

D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 317. [From N. J. A. Sloane, Nov 22 2009]

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Emma Y. Jin and Christian M. Reidys, Asymptotic Enumeration of RNA Structures with Pseudoknots, arXiv:0706.3137 [q-bio.BM], 2007, Theorem 5, p. 15.
Index entries for algebraic numbers, degree 2.

Equals 1+A090458. - R. J. Mathar, Aug 24 2008

Cf. A004253, A004254.

RealDigits[(5+Sqrt[21])/2,10,120][[1]] (* Harvey P. Dale, May 02 2011 *)

(sqrt(21)+5)/2 \\ Charles R Greathouse IV, Feb 11 2025

(4.791287...)^n = A090458 * A004254(n) + A004253(n). - Gary W. Adamson, Sep 11 2023
Equals lim_{n->oo} S(n, 5)/S(n-1, 5), with the S-Chebyshev polynomial (see A049310) S(n, 5) = A004254(n+1). - Wolfdieter Lang, Nov 15 2023
c^k = A004254(k)*c - A004254(k-1) for k >= 1, where c is the present constant. - Andrea Pinos, Jul 19 2024
Minimal polynomial: x^2 - 5*x + 1. - Stefano Spezia, Jul 02 2025

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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A090654 Decimal expansion of 4 + 2*sqrt(6).

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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).

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

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