A002163 -id:A002163 - cp's OEIS frontend

A208899 Decimal expansion of sqrt(5)/3.

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

Offset: 0

R. J. Mathar, Mar 03 2012

Equals the absolute value of the cosine of the dihedral angle between two adjacent faces of the regular icosahedron.

			0.7453559924...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, Maximum and Minimum Value of f(x)
Michael Penn, an "extreme" extreme value problem., YouTube video, 2022.
Wikipedia, Exsphere

Cf. A002163, A020837.

RealDigits[Sqrt[5]/3,10,120][[1]] (* Harvey P. Dale, Jul 03 2013 *)

first(n) = default(realprecision, max(28, n+10)); digits(((sqrt(5)/3)*10^n)\1) \\ David A. Corneth, Dec 19 2022

Equals A002163/3 = 20*A020837/3.

Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)/5^k. - Amiram Eldar, Aug 03 2020

A241149 Decimal expansion of sqrt(2) + sqrt(3) + sqrt(5).

Original entry on oeis.org

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

Offset: 1

K. D. Bajpai, Apr 16 2014

This is an algebraic integer, with its minimal polynomial being x^8 - 40x^6 + 352x^4 - 960x^2 + 576. - Alonso del Arte, Apr 17 2014

			5.382332347441762038738308734446846680953095488798854425503383962...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Penn, When showing square root 2 is irrational is too easy, YouTube video, 2022.

Cf. A002163 (decimal expansion: sqrt(5)).
Cf. A002193 (decimal expansion: sqrt(2)).
Cf. A002194 (decimal expansion: sqrt(3)).

Sqrt(2) + Sqrt(3) + Sqrt(5); // G. C. Greubel, Jul 27 2018

evalf(add(sqrt(ithprime(i)), i=1..3), 121);  # Alois P. Heinz, Jun 13 2022

RealDigits[Sqrt[2] + Sqrt[3] + Sqrt[5], 10, 200]

sqrt(2) + sqrt(3) + sqrt(5) \\ G. C. Greubel, Jul 27 2018

A265297 Decimal expansion of sum{x - c(2n-1), n=1,2,...}, where c = convergents to (x = sqrt(5)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.236844248570148187595380178229901194760980420...

Cf. A002163, A265298, A265299, A265288 (guide).

x = Sqrt[5]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265297 *)
RealDigits[s2, 10, 120][[1]]  (* A265298 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265299 *)

A265298 Decimal expansion of sum{c(2n) - x, n=1,2,...}, where c = convergents to (x = sqrt(5)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.01397529048416069463043193702846928135380025564...

Cf. A002163, A265297, A265299, A265288 (guide).

x = Sqrt[5]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265297 *)
RealDigits[s2, 10, 120][[1]]  (* A265298 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265299 *)

A265299 Decimal expansion of sum{c(2n) - c(2n-1), n=1,2,...}, where c = convergents to (x = sqrt(5)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.25081953905430888222581211525837047611...

Cf. A002163, A265297, A265298, A265288 (guide).

x = Sqrt[5]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265297 *)
RealDigits[s2, 10, 120][[1]]  (* A265298 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265299 *)

A305308 Decimal expansion of Lagrange(4) = sqrt(1517)/13.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jun 25 2018

For every irrational number alpha not equivalent to each of the following three numbers i) golden section A001622, ii) sqrt(2) = A002193 and iii) (5 + sqrt(221))/14 = A177841 there exist infinitely many rational numbers h/k (in lowest terms) such that |alpha - h/k| < 1/(Lagrange(4)*k^2). The constant L(4) cannot be replaced by a larger number because then the statement becomes false for, e.g., alpha = (23 + sqrt(1517))/26. Two real numbers x and y are equivalent if there exist integers p, q, r and s with |p*s - q*r| = 1 such that y = (p*x + q)/(r*x + s) (unimodular transformation). This means that the continued fractions of x and y become eventuakky identical.
See the references (in Havil, p. 174, equivalence classes of numbers should have been excluded).
The continued fraction of Lagrange(4) is [2; repeat(1, 252, 3, 1012, 3, 252, 1, 4)]. 1/L(4) = 0.3337725078... < 1/3.
Perron's numbers M(xi) (pp. 4, 5), for M(xi) < 3, are the Lagrange numbers sqrt(9*Q^2 - 4)/Q, with Q = Q(n) = A002559(n), n >= 1, and his corresponding xi(4) = (sqrt(1517) + 23)/26 with a purely periodic simple continued fraction [repeat(2, 2, 1, 1, 1, 1)].
Cassels (p. 18) uses the version: For irrational theta not equivalent to the above given three numbers i), ii) and iii) there are infinitely many solutions of q*||q*theta|| < 1/Lagrange(4), where 1/Lagrange(4) cannot be improved for theta equivalent to -29/26 + (1/26)*sqrt(1517). Here ||x|| is the positive difference between x and the nearest integer.

			2.9960526298692994692341394026263186397583021915005644481405263406560103404...

J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 174-175 and 221-224.
J. F. Koksma, Diophantische Approximationen, 1936, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vierter Band, Teil 4, Julius Springer, Berlin, pp. 29-33.
Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, p. 14.
Oskar Perron, Über die Approximation irrationaler Zahlen durch rationale, 4. Abhandlung, pp. 1- 17, and part II., 8. Abhandlung, pp. 1-12. Sitzungsber. Heidelberger Akademie der Wiss., 1921, Carls Winters Universitätsbuchhandlung.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 10. 6 B, pp. 312-314.
Jörn Steuding, Diophantine Analysis, 2005, Chapman & Hall/CRC, pp. 80-82.

Encyclopedia of Mathematics, Diophantine approximations.
A. A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann, 15 (18) (1879) 381-406.
A. A. Markoff, Sur les formes quadratiques binaires indéfinies (Second mémoire), Math. Ann, 17 (1880) 379-399.

The Lagrange numbers for n = 1..3 are A002163, A010466, A200991.

Cf. A001622, A002559.

RealDigits[Sqrt[1517]/13,10,120][[1]] (* Harvey P. Dale, Apr 12 2022 *)

Lagrange(4) = sqrt(9*M(4)^2 - 4)/M(4) = sqrt(9*13^2 - 4)/13 = sqrt(1517)/13, with the Markoff number M(4) = A002559(4) = 13.

A322159 Decimal expansion of 1 - 1/sqrt(5).

Original entry on oeis.org

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

Offset: 0

Tristan Cam, Nov 29 2018

Continued fraction: [0;1,1,4,4,4...].

Least root of the polynomial: 5x^2 - 10x + 4.

			0.552786404500042060718165266253744752911876328077...

Cf. A001622, A002163, A020762, A081010, A244847, A296182.

evalf[110](1-1/sqrt(5)); # Muniru A Asiru, Dec 01 2018

RealDigits[1-1/Sqrt[5], 10, 100][[1]] (* Amiram Eldar, Nov 29 2018 *)

Equals 1 - 1/A002163.
Equals 1/(1 - cos(4*Pi/5)) = (1/2)*csc(2*Pi/5)^2.
Also equal to 2/(phi*sqrt(5)) = 2/(A001622*A002163).
Equals 1 - A020762. - Andrew Howroyd, Nov 30 2018
From Amiram Eldar, Nov 28 2024: (Start)
Equals 2*A244847 = 1/A296182.
Equals Product_{k>=0} (1 - 1/A081010(k)). (End)

A349850 Decimal expansion of Sum_{k>=1} H(k)*F(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 02 2021

			3.96874800690391485217106364061998568869842436396224...

Hideyuki Ohtsuka, Problem B-1200, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 4 (2016), p. 367; Harmonic and Fiboancci [sic]/Lucas Numbers, Solution to Problem B-1200 by Kenny B. Davenport, ibid., Vol. 55, No. 4 (2017), pp. 372-373.

Cf. A000045, A001008, A001622, A002162, A002163, A002390, A002805, A349851.

RealDigits[2*Log[2] + 12*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals log(4*phi^(12/sqrt(5))) = 2*log(2) + 12*log(phi)/sqrt(5), where phi is the golden ratio (A001622).

A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 06 2024

			72.60725303413392187893153397383948620117264765443...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 8.

Cf. A377751 (volume), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A010527 (analogous for a regular icosahedron, with offset 1).
Cf. A002163, A002194, A375067.

First[RealDigits[3*(10*Sqrt[3] + Sqrt[25 + Sqrt[500]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "SurfaceArea"], 10, 100]]

3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) \\ Charles R Greathouse IV, Feb 05 2025

Equals 3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) = 30*A002194 + 3*sqrt(25 + 10*A002163).

Equals 30*(A002194 + A375067).

A377751 Decimal expansion of the volume of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 07 2024

			55.28773075812273923639861693886121953098664736582...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 2.

Cf. A377750 (surface area), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A102208 (analogous for a regular icosahedron).
Cf. A002163.

First[RealDigits[(125 + 43*Sqrt[5])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "Volume"], 10, 100]]

(125 + 43*sqrt(5))/4 \\ Charles R Greathouse IV, Feb 05 2025

Equals (125 + 43*sqrt(5))/4 = (125 + 43*A002163)/4.

cp's OEIS Frontend

A208899 Decimal expansion of sqrt(5)/3.

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A241149 Decimal expansion of sqrt(2) + sqrt(3) + sqrt(5).

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A265297 Decimal expansion of sum{x - c(2n-1), n=1,2,...}, where c = convergents to (x = sqrt(5)).

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A265298 Decimal expansion of sum{c(2n) - x, n=1,2,...}, where c = convergents to (x = sqrt(5)).

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A265299 Decimal expansion of sum{c(2n) - c(2n-1), n=1,2,...}, where c = convergents to (x = sqrt(5)).

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A305308 Decimal expansion of Lagrange(4) = sqrt(1517)/13.

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A322159 Decimal expansion of 1 - 1/sqrt(5).

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Maple

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A349850 Decimal expansion of Sum_{k>=1} H(k)*F(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

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