A020760 -id:A020760 - cp's OEIS frontend

A376870 Reduced numerators of Newton's iteration for 1/sqrt(3), starting with 1/3.

1, 4, 130, 2739685, 21055737501685791580, 9337539302589041654242365815942422114384262970589593842110

Offset: 0

Steven Finch, Oct 07 2024

An explicit formula for a(n) is not known, although it arises from a recurrence and the corresponding denominators are simply 3^((3^n + 1)/2) = 3*A134799(n).

Next term is too large to include.

			a(1) = 4 because b(1) = (3/2)*(1/3)*(1 - 1/9) = 4/9.
1/3, 4/9, 130/243, 2739685/4782969, ... = A376870(n)/(3*A134799(n)).

Paolo Xausa, Table of n, a(n) for n = 0..7
X. Gourdon and P. Sebah, Pythagoras' Constant.

Cf. A020760, A134799, A376867.

Module[{n = 0}, NestList[#*(3^(3^n++ + 1) - #^2)/2 &, 1, 6]] (* Paolo Xausa, Oct 17 2024 *)

from itertools import count, islice
def A376870_gen(): # generator of terms
    p = 1
    for k in count(0):
        yield p
        p = p*(3**(3**k+1)-p**2)>>1
A376870_list = list(islice(A376870_gen(),6)) # Chai Wah Wu, Oct 11 2024

a(n) is the reduced numerator of b(n) = (3/2)*b(n-1)*(1 - b(n-1)^2); b(0) = 1/3.
Limit_{n -> oo} a(n)/(3*A134799(n)) = 1/sqrt(3) = A020760.
a(n+1) = a(n)*(3^(3^n+1)-a(n)^2)/2. - Chai Wah Wu, Oct 11 2024

A134799 a(n) = 3^((3^n - 1)/2).

Original entry on oeis.org

1, 3, 81, 1594323, 12157665459056928801, 5391030899743293631239539488528815119194426882613553319203

Offset: 0

Yasutoshi Kohmoto, Jan 09 2008

Number of partitions into "bus routes" of the graph G_{n+1} defined below.

These seem to be one-third the reduced denominators of Newton's iteration for 1/sqrt(3), starting with 1/3. - Steven Finch, Oct 08 2024

			.........|..................G_1
****
.......__|__................G_2
.........|
****
.__|_____|_____|__..........G_3
...|.....|.....|
.........|
.......__|__
.........|
****.
..._|_........._|_..........G_4
_|__|_____|_____|__|_
.|._|_....|...._|_.|
....|.....|.....|
......_|__|__|_
.......|._|_.|
..........|
****
G_1 = o---. = rooted tree with one edge and one leaf node. For n > 0, G_{n+1} is obtained from G_n by splitting each leaf node into three.

Andrew Howroyd, Table of n, a(n) for n = 0..7
Author?, Mitsumata tree
X. Gourdon and P. Sebah, Pythagoras' Constant.

Cf. A131709, A020760, A376870.

3^((3^Range[0, 6] - 1)/2) (* Paolo Xausa, Oct 17 2024 *)

a(n) is conjectured to be one-third the reduced denominator of b(n) = (3/2)*b(n-1)*(1 - b(n-1)^2); b(0) = 1/3. - Steven Finch, Oct 08 2024

Limit_{n -> oo} A376870(n)/(3*a(n)) = 1/sqrt(3) = A020760. - Steven Finch, Oct 08 2024

Edited by N. J. A. Sloane, Jan 29 2008

a(5) from Andrew Howroyd, Oct 07 2024

A156309 Decimal expansion of the absolute value of the larger solution of (n^2+n)/2 = -1/12. (Real root q of 6n^2 + 6n + 1, the other root being p = -1-q.)

Original entry on oeis.org

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

Offset: 0

Daniele P. Morelli, Feb 07 2009

The formula returning the n-th triangular number (A000217) is (n^2+n)/2. On the other hand, Ramanujan's identity claims that the value of the infinite sum 1+2+3+.... is -1/12. This irrational number is the solution of the equation (n^2+n)/2 = -1/12, that is, the "limit" triangular number.
Equals the Knuth's random generators constant, that is, the ratio c/m in congruence random number generators of the type X_(n+1) = (aX_n +c) mod (m) which minimizes the correlation between successive values. - Stanislav Sykora, Nov 13 2013
It is also the fraction of the full solid angle cut out by a cone having the magic angle (A195696) as its polar angle. - Stanislav Sykora, Nov 13 2013

			The two roots of 6n^2 + 6n + 1 = 0 are -0.21132... and -0.78867513... (Cf. A020769.)

B. Candelpergher, Ramanujan summation of divergent series. Lectures notes in mathematics 2185, Springer 2017.
D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, 1969, Chapter 3.3.3.

P. J. Cameron and V. Yildiz, Counting false entries in truth tables of bracketed formulas connected by implication, arXiv:1106.4443 [math.CO], 2011.
Michael Penn, What is the radius of 🔴 ?, YouTube video, 2021.

Cf. A000217, A020760, A020769, A165663, A195696, A334843.

First[RealDigits[(3 - Sqrt[3])/6, 10, 100]] (* Paolo Xausa, Jun 25 2024 *)

abs(solve(n=-1/2, 0, 6*n^2+6*n+1)) \\ Michel Marcus, Oct 05 2013

(1 - 1/sqrt(3))/2 = (1 - A020760)/2 = 1/2 - A020769. - R. J. Mathar, Feb 10 2009
Equals - HurwitzZeta(-1, (9 - sqrt(3))/6). - Peter Luschny, Jul 05 2020
Equals (3 - sqrt(3))/6. - Michel Marcus, Jun 10 2021
Equals 1/A165663 = A334843/3. - Hugo Pfoertner, Jun 25 2024

Flipped sign of definition, corrected offset, simplified formula R. J. Mathar, Feb 10 2009

A212886 Decimal expansion of 2/(3sqrt(3)) = 2sqrt(3)/9.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, May 29 2012

Consider any cubic polynomial f(x) = a(x - r)(x - (r + s))(x -(r + 2s)), where a, r, and s are real numbers with s > 0 and nonzero a; i.e., any cubic polynomial with three distinct real roots, of which the middle root, r + s, is equidistant (with distance s) from the other two. Then the absolute value of f's local extrema is |a|*s^3*(2*sqrt(3)/9). They occur at x = r + s +- s*(sqrt(3)/3), with the local maximum, M, at r + s - s*sqrt(3)/3 when a is positive and at r + s + s*sqrt(3)/3 when a is negative (and the local minimum, m, vice versa). Of course m = -M < 0.
A quadratic number with denominator 9 and minimal polynomial 27x^2 - 4. - Charles R Greathouse IV, Apr 21 2016
This constant is also the maximum curvature of the exponential curve, occurring at the point M of coordinates [x_M = -log(2)/2 = (-1/10)*A016655; y_M = sqrt(2)/2 = A010503]. The corresponding minimum radius of curvature is (3*sqrt(3))/2 = A104956 (see the reference Eric Billault and the link MathStackExchange). - Bernard Schott, Feb 02 2020

			0.384900179459750509672765853667971637098401167513417917345734...

Eric Billault, Walter Damin, Robert Ferréol et al., MPSI - Classes Prépas, Khôlles de Maths, Ellipses, 2012, exercice 17.07 pages 386, 389-390.

MathStackExchange, Apex of an Exponential Function, 2015.
Index entries for algebraic numbers, degree 2

Cf. A002194, A104956, A020760.

Cf. A010503, A016655.

RealDigits[2/(3*Sqrt[3]), 10, 105] (* T. D. Noe, May 31 2012 *)

default(realprecision, 1000); 2*sqrt(3)/9

(2/9)*sqrt(3) = (2/9)*A002194.

A343057 Decimal expansion of tan(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.098491403357164253077197...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8.

Cf. A343055 (sin(Pi/32)), A343056 (cos(Pi/32)).

tan(Pi/m): A002194 (m=3), A019934 (m=5), A020760 (m=6), A343058 (m=7), A188582 (m=8), A019918 (m=9), A019916 (m=10), A019913 (m=12), A343059 (m=14), A019910 (m=15), A343060 (m=16), A343061 (m=17), A019908 (m=18), A019907 (m=20), A343062 (m=24), A019904 (m=30), A343057 (m=32), A019903 (m=36).

R:= RealField(125); Tan(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Tan[Pi/32], 10, 120, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/32)
```

sqrt((2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))))

numerical_approx(tan(pi/32), digits=125) # G. C. Greubel, Sep 30 2022

Equals sqrt( (2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))) ).

Equals A232738/(1+A232737). - R. J. Mathar, Sep 06 2025

A359837 Decimal expansion of the unsigned ratio of similitude between an equilateral reference triangle and its first Morley triangle.

Original entry on oeis.org

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

Offset: 0

Frank M Jackson, Jan 14 2023

The first Morley triangle of any reference triangle is always equilateral. Therefore a reference equilateral triangle and its first Morley triangle will be in a homothetic relationship. This sequence is the real terms of a constant that is the ratio of similitude of the homothety. The constant is negative.

If an equilateral triangle has a side a, a circumradius R and a first Morley triangle with side a', then a = R*sqrt(3) and a' = 8*R*(sin(Pi/9))^3, so the ratio of similitude between the two triangles is a'/a = (8/sqrt(3)) * (sin(Pi/9))^3. - Bernard Schott, Feb 06 2023

			0.1847925309040953727013520475722037558709135597926517172356...

Wikipedia, Morley's trisector theorem.
Index entries for algebraic numbers, degree 3.

Cf. A019819, A019829, A019879, A020760.

RealDigits[Sin[Pi/18]/Cos[Pi/9], 10, 100][[1]]
N[Solve[x^3 + 3*x^2 - 6*x + 1 == 0, {x}][[2]], 90]

sin(Pi/18)/cos(Pi/9) \\ Michel Marcus, Jan 15 2023

Equals sin(Pi/18)/cos(Pi/9).
A root of x^3+3*x^2-6*x+1.
Equals A019819/A019879. - Michel Marcus, Jan 15 2023
Equals 8 * A020760 * A019829^3. - Bernard Schott, Feb 06 2023

A184977 a(n) = Sum_{k=1..n} floor(k*gamma) where gamma is Euler's constant (A001620).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 13, 17, 22, 27, 33, 39, 46, 54, 62, 71, 80, 90, 100, 111, 123, 135, 148, 161, 175, 190, 205, 221, 237, 254, 271, 289, 308, 327, 347, 367, 388, 409, 431, 454, 477, 501, 525, 550, 575, 601, 628, 655, 683, 711, 740, 770, 800, 831, 862, 894, 926, 959, 993, 1027, 1062

Offset: 1

Michel Lagneau, Mar 27 2011

a(n) = A183143(n) for n = 1..96 where A183143(n) is the sequence floor(1/r) + floor(2/r) + ... + floor(n/r) and r = sqrt(3). It is interesting to note that a(n)/n^2 converges to gamma/2.
gamma = 0.57721566490153286060651209... (A002852)
1/sqrt(3) = 0.577350269189625764509148... (A020760)
Starts to differ from A183143 at a(97). - R. J. Mathar, Aug 28 2025

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001620, A038128.

R:=RealField(100); [(&+[Floor(k*EulerGamma(R)): k in [1..n]]): n in [1..50]]; // G. C. Greubel, Aug 27 2018

with(numtheory):Digits:=500:s:=0:c:=evalf(gamma(0)):for n from 1 to 100 do:
  s:=s+floor(n*c):printf(`%d, `,s):od:

Table[Sum[Floor[k*EulerGamma], {k, 1, n}], {n, 50}] (* G. C. Greubel, Jun 02 2017 *)

a(n) = sum(k=1, n, floor(k*Euler)); \\ Michel Marcus, Apr 02 2017

Partial sums of A038128.

Name edited by Jon E. Schoenfield, Apr 02 2017

A263574 Beatty sequence for 1/sqrt(3) - log(phi)/3575 where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43

Offset: 0

Karl V. Keller, Jr., Oct 21 2015

nonn

The number 1/sqrt(3) - log(phi)/3575 (=0.577215664483...) is an approximation to Euler's constant (A001620) (=0.577215664901...).

M. Hudson found a similar Euler-Mascheroni constant approximation (see link), 1/sqrt(3)-1/7429 (=0.57721566157...).

			For n=9, floor(9*(0.577215664483)) = floor(5.194940980347) = 5.

Karl V. Keller, Jr., Table of n, a(n) for n = 0..100000
Xavier Gourdon and Pascal Sebah, Collection of formulas for Euler's constant,Euler's constant.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant Approximations.

Cf. A001620, A020760 (1/sqrt(3)), A038128 (Beatty sequence for Euler's constant), A097337 (Beatty sequence for 1/sqrt(3)).

phi:= (1+Sqrt(5))/2; [Floor(n*(1/Sqrt(3) - Log(phi)/3575)): n in [0..100]]; // G. C. Greubel, Sep 05 2018

Table[Floor[n (1/Sqrt@ 3 - Log[GoldenRatio]/3575)], {n, 0, 75}] (* Michael De Vlieger, Nov 12 2015 *)

{phi = (1+sqrt(5))/2}; vector(100, n, n--; floor(n*(1/sqrt(3) - log(phi)/3575))) \\ G. C. Greubel, Sep 05 2018

from sympy import floor, log, sqrt
for n in range(0,101):print(floor(n*(1/sqrt(3)-log(1/2+sqrt(5)/2)/3575)),end=', ')

a(n) = floor(n*(1/sqrt(3) - log(phi)/3575)).

a(n) = A038128(n) for n < 58628.

A270230 Decimal expansion of 3/(4*Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Mar 13 2016

Consider generic prisms with triangular bases (tp), enclosed by a sphere, and let f(tp) be the fraction of the sphere volume occupied by any of them (i.e., the ratio of the prism volume to the sphere volume). Then this constant is the supremum of f(tp). It is attained by prisms which have as their base equilateral triangles with edge lengths r*sqrt(2), and rectangular side faces that are r*sqrt(2) wide and r*2/sqrt(3) high, where r is the radius of the enclosing, circumscribed sphere.
An intriguing fact is that the volume of such a best-fitting prism is exactly r^3. Hence, 1/a is the volume of a sphere with radius 1.
Examples of similar constants obtained for other shapes enclosed by spheres are: A020760 for cylinders and A165952 for cuboids.

			0.238732414637843003653325645058771543051689468610684673121501016...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Index entries for transcendental numbers

Cf. A002193, A019699 (one tenth of 1/a), A020760, A020832 (one tenth of 2/sqrt(3)), A165952.

First@ RealDigits[N[3/4/Pi, 120]] (* Michael De Vlieger, Mar 15 2016 *)

PARI
```
3/4/Pi
```

A270395 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r(k) = 1/Fibonacci(k+1).

Original entry on oeis.org

2, 7, 57, 2713, 4918440, 22223269372702, 355194969748884199331083933, 896996605353313749663062291034129550464167047150212163, 710754225314643793883316602476833806192189702887005360976366457324682443530843343068467237316280025378530303

Offset: 1

Clark Kimberling, Mar 22 2016

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ..., the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

			sqrt(1/3) = 1/2 + 1/(2*7) + 1/(3*57) + 1/(5*2713) + ...

Clark Kimberling, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A000045, A020760.

r[k_] := 1/Fibonacci[k+1]; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[1/3]; Table[n[x, k], {k, 1, z}]

r(k) = 1/fibonacci(k+1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016

cp's OEIS Frontend

A376870 Reduced numerators of Newton's iteration for 1/sqrt(3), starting with 1/3.

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A134799 a(n) = 3^((3^n - 1)/2).

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A156309 Decimal expansion of the absolute value of the larger solution of (n^2+n)/2 = -1/12. (Real root q of 6n^2 + 6n + 1, the other root being p = -1-q.)

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A212886 Decimal expansion of 2/(3*sqrt(3)) = 2*sqrt(3)/9.

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A343057 Decimal expansion of tan(Pi/32).

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A359837 Decimal expansion of the unsigned ratio of similitude between an equilateral reference triangle and its first Morley triangle.

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A184977 a(n) = Sum_{k=1..n} floor(k*gamma) where gamma is Euler's constant (A001620).

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A212886 Decimal expansion of 2/(3sqrt(3)) = 2sqrt(3)/9.