A188640 -id:A188640 - cp's OEIS frontend

A098317 Decimal expansion of phi^3 = 2 + sqrt(5).

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

Offset: 1

Eric W. Weisstein, Sep 02 2004

This sequence is also the decimal expansion of ((1+sqrt(5))/2)^3. - Mohammad K. Azarian, Apr 14 2008
This is the length/width ratio of a 4-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 10 2011
Its continued fraction is [4, 4, ...] (see A010709). - Robert G. Wilson v, Apr 10 2011

			4.23606797749978969640917366873127623544061835961152572427...

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
Alexey Stakhov, The mathematics of harmony: from Euclid to contemporary mathematics and computer science, World Scientific, Singapore, 2009, p. 657.

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

Cf. A001622, A014176, A098316, A098318.

Cf. A001076, A049310.

SetDefaultRealField(RealField(100)); 2+Sqrt(5); // G. C. Greubel, Jun 30 2019

RealDigits[N[2+Sqrt[5],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

sqrt(5)+2 \\ Charles R Greathouse IV, Mar 11 2012

numerical_approx(2+sqrt(5), digits=100) # G. C. Greubel, Jun 30 2019

2 plus the constant in A002163. - R. J. Mathar, Sep 02 2008
Equals 3 + 4*sin(Pi/10) = 1 + 4*cos(Pi/5) = 1 + 4*sin(3*Pi/10) = 3 + 4*cos(2*Pi/5) = 1 + csc(Pi/10). - Arkadiusz Wesolowski, Mar 11 2012
Equals lim_{n -> infinity} F(n+3)/F(n) = lim_{n -> infinity} (1 + 2*F(n+1)/F(n)) = 2 + sqrt(5), with F(n) = A000045(n). - Arkadiusz Wesolowski, Mar 11 2012
Equals exp(arcsinh(2)), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals Sum_{n>=1} n/phi^n = phi/(phi-1)^2 = phi^3. - Richard R. Forberg, Jun 29 2014
Equals 1 + 2*phi, with phi = A001622, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Dec 10 2022
c^n = A001076(n-1) + c * A001076(n); where c = 2 + sqrt(5). - Gary W. Adamson, Oct 09 2023
Equals lim_{n -> infinity} = S(n, 2*(-1 + 2*phi))/S(n-1, 2*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). See also the above limit formula with Fibonacci numbers. - Wolfdieter Lang, Nov 15 2023

Title expanded to include observation from Mohammad K. Azarian by Charles R Greathouse IV, Mar 11 2012

A098316 Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 02 2004

For reasons following from the formula section, this constant could be called "the bronze ratio". For this, compare with A001622 and A014176.
If c is this constant and n > 0, then for n even, c^n = [A100230(n), 1, A100230(n)-1, 1, A100230(n)-1, 1, A100230(n)-1, 1, ...], for n odd, c^n = [A100230(n)+1, A100230(n)+1, A100230(n)+1, ...]. - Gerald McGarvey, Dec 15 2007
This is the shape of a 3-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 10 2011
From Vladimir Shevelev, Mar 02 2013: (Start)
An analog of Fermat theorem: for prime p, round(c^p) == 3 (mod p).
A generalization for "metallic" constants c_N = (N+sqrt(N^2+4))/2, N>=1: for prime p, round((c_N)^p) == N (mod p). (End)
This is the positive real algebraic number c of degree 2 with minimal polynomial x^3 - x - 1. The other negative root is 3 - c. - Wolfdieter Lang, Aug 29 2022
c^n = c*A006190(n) + A006190(n-1). - Gary W. Adamson, Apr 02 2024

			3.30277563...

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

Cf. A001622, A014176, A098317, A098318, A000032, A006497, A080039, A049310.

RealDigits[(3 + Sqrt[13])/2, 10, 100][[1]] (* G. C. Greubel, Apr 16 2017 *)

(3 + sqrt(13))/2 \\ Charles R Greathouse IV, Jul 24 2013

3 plus the constant in A085550. - R. J. Mathar, Sep 02 2008
From Hieronymus Fischer, Jan 02 2009: (Start)
Set c:=(3+sqrt(13))/2. Then the fractional part of c^n equals 1/c^n, if n odd. For even n, the fractional part of c^n is equal to 1-(1/c^n).
c:=(3+sqrt(13))/2 satisfies c-c^(-1)=floor(c)=3, hence c^n + (-c)^(-n) = round(c^n) for n>0, which follows from the general formula of A001622.
1/c=(sqrt(13)-3)/2.
See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).
Other examples of constants x satisfying the relation x-x^(-1)=floor(x) include A001622 (the golden ratio: where floor(x)=1) and A014176 (the silver ratio: where floor(x)=2). (End)
c=3+sum{k>=1}(-1)^(k-1)/(A006190(k)*A006190(k+1)). - Vladimir Shevelev, Feb 23 2013
A generalization for "metallic" constants c_N = (N+sqrt(N^2+4))/2, N>=1. Let {A_N(n), n>=0} be the sequence 0, 1, N, N^2+1, N^3+2*N, N^4+3*N^2+1,..., a(N) = N*a(N-1) + a(N-2). Then c_N = N + sum_{n>=1} (-1)^(n-1)/(A_N(n)*A_N(n+1)) (cf. A001622, A014176, A098316, A098317, A098318). - Vladimir Shevelev, Feb 23 2013
Equals lim_{n->oo} S(n, sqrt(13))/S(n-1, sqrt(13)), with the S-Chebyshev polynomial (see A049310). - Wolfdieter Lang, Nov 15 2023

A060997 Decimal representation of continued fraction 1, 2, 3, 4, 5, 6, 7, ...

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, May 14 2001

The value of this continued fraction is the ratio of two Bessel functions: BesselI(0,2)/BesselI(1,2) = A070910/A096789. Or, equivalently, to the ratio of the sums: Sum_{n>=0} 1/(n!n!) and Sum_{n>=0} n/(n!n!). - Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

1.43312...=[1,2,3,4,5,...] = shape of a rectangle which partitions into n squares at stage n; i.e., this is an example of the match between the continued fraction of a number r and a rectangle having shape r. See A188640. - Clark Kimberling, Apr 09 2011

			1.433127426722311758317183455775...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000 (corrected by _Sean A. Irvine_, Apr 29 2022)
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]

Cf. A052119, A001053.

With[{nn = 110}, RealDigits[FromContinuedFraction[Range[nn]], 10, nn][[1]]]
(* Or *) RealDigits[ BesselI[0, 2] / BesselI[1, 2], 10, 110] [[1]]
(* Or *) RealDigits[ Sum[1/(n!n!), {n, 0, Infinity}] / Sum[n/(n!n!), {n, 0, Infinity}], 10, 110] [[1]]

set_display('none)$fpprec:100$bfloat(cfdisrep(makelist(x,x,1,1000))); /* Dimitri Papadopoulos, Oct 25 2022 */

besseli(0,2)/besseli(1,2) \\ Charles R Greathouse IV, Feb 19 2014

1/A052119.

A188738 Decimal expansion of e-sqrt(e^2-1).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Apr 11 2011

Decimal expansion of the shape of a lesser 2e-contraction rectangle.
The shape of a rectangle WXYZ, denoted by [WXYZ], is defined by length/width: [WXYZ]=max{|WX|/|YZ|, |YZ|/|WX|}.  Consider the following configuration of rectangles AEFD, EBCF, ABCD, where AEFD is not a square:
D................F....C
.......................
.......................
.......................
A................E....B
Suppose that ABCD is given and that the shape r=[ABCD] exceeds 2.  The "r-contraction rectangles" of ABCD are here introduced as the rectangles AEFD and EBCF for which [AEFD]=[EBCF] and |AE|<>|EB|.  That is, ABCD has the prescribed shape r, and AEFD and EBCF are mutually similar without being congruent.  It is easy to prove that [AEFD]=(1/2)(r-sqrt(-4+r^2)) or [AEFD]=(1/2)(r+sqrt(-4+r^2)); in the former case, we call AEFD the "lesser r-contraction rectangle", and the latter, the "greater r-contraction rectangle".
Both r-contraction rectangles match the continued fraction of [AEFD] in the following way.  Write the continued fraction as [a(1),a(2),a(3),...].  Then, in the manner in which the continued fraction [1,1,1,...] matches the step-by-step removal of single squares from a golden triangle (as well as the manner in which the continued fraction [2,2,2,...] matches the step-by-step removal of 2 squares at a time from a silver triangle, etc.), remove a(1) squares at step 1, then remove a(2) squares at step 2, and so on, obtaining in the limit a partition of AEFD as an infinite set of squares.
For (related) r-extension rectangles, see A188640.

			0.190623604147330614259428256541555268663022202.. = 1/A188739 with continued fraction 0, 5, 4, 15, 6, 1, 13, 2, 1, 1, 21, 3, 2, 16, 1, 4, 1, 1, 157,...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.
Index entries for transcendental numbers

Cf. A001113, A188739 (inverse), A188627 (continued fraction), A188640.

SetDefaultRealField(RealField(100)); Exp(1) - Sqrt(Exp(2)-1); // G. C. Greubel, Nov 01 2018

evalf(exp(1)-sqrt(exp(2)-1),140); # Muniru A Asiru, Nov 01 2018

r = 2 E; t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

default(realprecision, 100); exp(1) - sqrt(exp(2)-1) \\ G. C. Greubel, Nov 01 2018

A098318 Decimal expansion of [5, 5, ...] = (5 + sqrt(29))/2.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 02 2004

The "metallic" constants A001622, A014176 etc. are defined inserting a = 1, 2, 3, 4, ... into (a+sqrt(a^2+4))/2. [Stakhov & Aranson] - R. J. Mathar, Feb 14 2011

This is the length/width ratio of a 5-extension rectangle; see A188640 where the metallic constants are defined for rational numbers. - Clark Kimberling, Apr 09 2011

			5.19258240356725201562535524577016477814756...

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. Stakhov and Samuil Aranson, Hyperbolic Fibonacci and Lucas functions, Golden Fibonacci Goniometry, Bodnar's Geometry, ..., Appl. Math. 2 (1) (2011) 74-84.
Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A001622, A014176, A098316, A098317, A010716 (continued fraction).

Cf. A052918, A049310.

SetDefaultRealField(RealField(100)); (5 + Sqrt(29))/2; // G. C. Greubel, Jun 30 2019

r=5; t=(r+(4+r^2)^(1/2))/2; FullSimplify[t]
N[t,130]
RealDigits[N[t,130]][[1]]
ContinuedFraction[t,120] (* Clark Kimberling, Apr 09 2011 *)

(5 + sqrt(29))/2 \\ Charles R Greathouse IV, Jul 24 2013

numerical_approx((5+sqrt(29))/2, digits=100) # G. C. Greubel, Jun 30 2019

5 plus the constant in A085551. - R. J. Mathar, Sep 02 2008
c^n = A052918(n-2) + A052918(n-1) * c, where c = (5 + sqrt(29))/2. - Gary W. Adamson, Oct 09 2023
Equals lim_{n->infinity} S(n, sqrt(29))/ S(n-1, sqrt(29)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A176398 Decimal expansion of 3+sqrt(10).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 16 2010

Continued fraction expansion of 3+sqrt(10) is A010722.
This is the shape of a 6-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011
c^n = c*A005668(n) + A005668(n-1). - Gary W. Adamson, Apr 04 2024

			6.16227766016837933199...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Wikipedia, Metallic mean.
Index entries for algebraic numbers, degree 2

Cf. A010467 (decimal expansion of sqrt(10)), A010722 (all 6's sequence).

Cf. A049310.

Digits:=100; evalf(3+sqrt(10)); # Wesley Ivan Hurt, Mar 07 2014

r=6; t=(r+(4+r^2)^(1/2))/2; RealDigits[N[t,130]][[1]] (* Clark Kimberling, Apr 09 2011 *)

3+sqrt(10) \\ Charles R Greathouse IV, Jul 24 2013

a(n) = A010467(n) for n >= 2.
Equals exp(arcsinh(3)), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals lim_{n->oo} S(n, 2*sqrt(10))/ S(n-1, 2*sqrt(10)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A188887 Decimal expansion of sqrt(2 + sqrt(3)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 12 2011

Decimal expansion of the length/width ratio of a sqrt(2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A sqrt(2)-extension rectangle matches the continued fraction [1,1,13,1,2,15,10,1,18,1,1,21,,...] (A188888) for the shape L/W = sqrt(2 + sqrt(3)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...].  Specifically, for the sqrt(2)-extension rectangle, 1 square is removed first, then 1 square, then 13 squares, then 1 square, ..., so that the original rectangle of shape sqrt(2 + sqrt(3)) is partitioned into an infinite collection of squares.
sqrt(2 + sqrt(3)) is also the shape of the greater sqrt(6)-contraction rectangle; see A188738.
This constant is also the length of the Steiner span of three vertices of a unit square. - Jean-François Alcover, May 22 2014
It is also the larger positive coordinate of (symmetrical) intersection points created by x^2 + y^2 = 4 circle and y = 1/x hyperbola. The smaller coordinate is A101263. - Leszek Lezniak, Sep 18 2018
Length of the shortest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Nov 12 2020

			1.931851652578136573499486399457794735267809678016809...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Burkard Polster, Irrational roots, Mathologer video (2018).
Index entries for algebraic numbers, degree 4.

Cf. A019884, A019973, A091998, A101263, A188640, A188888, A214726, A217870, A329247.

Sqrt(2 + Sqrt(3)); // G. C. Greubel, Apr 10 2018

r = 2^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[Sqrt[2 + Sqrt[3]], 10, 100][[1]] (* G. C. Greubel, Apr 10 2018 *)

sqrt(2 + sqrt(3)) \\ G. C. Greubel, Apr 10 2018

Equals (sqrt(6) + sqrt(2))/2.
Equals exp(asinh(cos(Pi/4))). - Geoffrey Caveney, Apr 23 2014
Equals cos(Pi/4) + sqrt(1 + cos(Pi/4)^2). - Geoffrey Caveney, Apr 23 2014
Equals i^(1/6) + i^(-1/6). - Gary W. Adamson, Jul 07 2022
Equals the largest root of x - 1/x = sqrt(2) and of x^2 + 1/x^2 = 4. - Gary W. Adamson, Jun 12 2023
Equals Product_{k>=0} ((12*k + 2)*(12*k + 10))/((12*k + 1)*(12*k + 11)). - Antonio Graciá Llorente, Feb 24 2024
From Amiram Eldar, Nov 23 2024: (Start)
Equals A214726 / 2 = 2 * A019884 = 1 / A101263 = exp(A329247) = A217870^2 = sqrt(A019973).
Equals Product_{k>=1} (1 - (-1)^k/A091998(k)). (End)

A135611 Decimal expansion of sqrt(2) + sqrt(3).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 03 2008

From Alexander R. Povolotsky, Mar 04 2008: (Start)
The value of sqrt(2) + sqrt(3) ~= 3.146264369941972342329135... is "close" to Pi. [See Borel 1926. - Charles R Greathouse IV, Apr 26 2014] We can get a better approximation by solving the equation: (2-x)^(1/(2+x)) + (3-x)^(1/(2+x)) = Pi.
Olivier Gérard finds that x is 0.00343476569746030039595770020414255107204742044644777... (End)
Another approximation to Pi is (203*sqrt(2)+ 197*sqrt(3))/200 = 3.1414968... - Alexander R. Povolotsky, Mar 22 2008
Shape of a sqrt(8)-extension rectangle; see A188640.  - Clark Kimberling, Apr 13 2011
This number is irrational, as instinct would indicate. Niven (1961) gives a proof of irrationality that requires first proving that sqrt(6) is irrational. - Alonso del Arte, Dec 07 2012
An algebraic integer of degree 4: largest root of x^4 - 10*x^2 + 1. - Charles R Greathouse IV, Sep 13 2013
Karl Popper considers whether this approximation to Pi might have been known to Plato, or even conjectured to be exact. - Charles R Greathouse IV, Apr 26 2014

			3.14626436994197234232913506571557044551247712918732870...

Emile Borel, Space and Time (1926).
Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (1961): 44.
Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): 44.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Susan Landau, Simplification of nested radicals, SIAM Journal on Computing 21.1 (1992): 85-110. See page 85. [Do not confuse this paper with the short FOCS conference paper with the same title, which is only a few pages long.]
Burkard Polster, Irrational roots, Mathologer video (2018)
Karl Popper, The Open Society and Its Enemies, 1962.
Index entries for algebraic numbers, degree 4

Cf. A188640, A089078, A002193, A002194, A010464, A340616.

SetDefaultRealField(RealField(100)); Sqrt(2) + Sqrt(3); // G. C. Greubel, Nov 20 2018

evalf(add(sqrt(ithprime(i)), i=1..2), 118);  # Alois P. Heinz, Jun 13 2022

r = 8^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A135611 *)
ContinuedFraction[t, 120]  (* A089078 *)
RealDigits[Sqrt[2] + Sqrt[3], 10, 100][[1]] (* G. C. Greubel, Oct 22 2016 *)

sqrt(2)+sqrt(3) \\ Charles R Greathouse IV, Sep 13 2013

numerical_approx(sqrt(2)+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

Sqrt(2)+sqrt(3) = sqrt(5+2*sqrt(6)). [Landau, p. 85] - N. J. A. Sloane, Aug 27 2018
Equals 1/A340616. - Hugo Pfoertner, May 08 2024
Equals Product_{k>=0} (((4*k + 1)*(12*k + 11))/((4*k + 3)*(12*k + 1)))^(-1)^k. - Antonio Graciá Llorente, May 22 2024

A176439 Decimal expansion of (7+sqrt(53))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (7+sqrt(53))/2 is A010727.
This is the shape of a 7-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011
c^n = c * A054413(n-1) + A054413(n-2), where c = (7+sqrt(53))/2. - Gary W. Adamson, Apr 14 2024

			(7+sqrt(53))/2 = 7.14005494464025913554...

Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A010506 (decimal expansion of sqrt(53)), A010727 (all 7's sequence).

Cf. A049310.

r=7; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(7+Sqrt[53])/2,10,120][[1]] (* Harvey P. Dale, Nov 03 2024 *)

(7+sqrt(53))/2 \\ Charles R Greathouse IV, Jul 24 2013

Equals lim_{n->oo} S(n, sqrt(53))/S(n-1, sqrt(53)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

Positive solution of x^2 - 7*x - 1 = 0. - Hugo Pfoertner, Apr 14 2024

A176458 Decimal expansion of 4+sqrt(17).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of 4+sqrt(17) is A010731.

This is the shape of an 8-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011

			4+sqrt(17) = 8.12310562561766054982...

Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A010473 (decimal expansion of sqrt(17)), A010731 (all 8's sequence).

Cf. A049310.

r=8; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]

4+sqrt(17) \\ Charles R Greathouse IV, Jul 24 2013

a(n) = A010473(n) for n > 1.
Equals exp(arcsinh(4)), since arcsinh(x)=log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals lim_{n->infinity} S(n, 2*sqrt(17))/S(n-1, 2*sqrt(17)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

cp's OEIS Frontend

A098317 Decimal expansion of phi^3 = 2 + sqrt(5).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A098316 Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A060997 Decimal representation of continued fraction 1, 2, 3, 4, 5, 6, 7, ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

A188738 Decimal expansion of e-sqrt(e^2-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A098318 Decimal expansion of [5, 5, ...] = (5 + sqrt(29))/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A176398 Decimal expansion of 3+sqrt(10).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica